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5.1: Roots and Radicals - Mathematics


Learning Objectives

  • Identify and evaluate square and cube roots.
  • Determine the domain of functions involving square and cube roots.
  • Evaluate (n)th roots.
  • Simplify radicals using the product and quotient rules for radicals.

Square and Cube Roots

Recall that a square root1 of a number is a number that when multiplied by itself yields the original number. For example, (5) is a square root of (25), because (5^{2} = 25). Since ((−5)^{2} = 25), we can say that (−5) is a square root of (25) as well. Every positive real number has two square roots, one positive and one negative. For this reason, we use the radical sign (√) to denote the principal (nonnegative) square root2 and a negative sign in front of the radical (−√) to denote the negative square root.

Zero is the only real number with one square root.

(sqrt { 0 } = 0 ext { because } 0 ^ { 2 } = 0)

Example (PageIndex{1}):

Evaluate.

  1. (sqrt { 121 })
  2. (- sqrt { 81 })

Solution

  1. (sqrt { 121 } = sqrt { 11 ^ { 2 } } = 11)
  2. (- sqrt { 81 } = - sqrt { 9 ^ { 2 } } = - 9)

If the radicand3, the number inside the radical sign, can be factored as the square of another number, then the square root of the number is apparent. In this case, we have the following property:

(sqrt { a ^ { 2 } } = a quad ext { if } quad a geq 0)

Or more generally,

(sqrt { a ^ { 2 } } = | a | quad ext { if } quad a in R)

The absolute value is important because (a) may be a negative number and the radical sign denotes the principal square root. For example,

Make use of the absolute value to ensure a positive result.

Example (PageIndex{2}):

Simplify: (sqrt { ( x - 2 ) ^ { 2 } }).

Solution

Here the variable expression (x − 2) could be negative, zero, or positive. Since the sign depends on the unknown quantity (x), we must ensure that we obtain the principal square root by making use of the absolute value.

Answer:

(| x - 2 |)

The importance of the use of the absolute value in the previous example is apparent when we evaluate using values that make the radicand negative. For example, when (x = 1),

Next, consider the square root of a negative number. To determine the square root of (−25), you must find a number that when squared results in (−25):

However, any real number squared always results in a positive number. The square root of a negative number is currently left undefined. For now, we will state that (sqrt { - 25 }) is not a real number. Therefore, the square root function4 given by (f ( x ) = sqrt { x }) is not defined to be a real number if the (x)-values are negative. The smallest value in the domain is zero. For example,(f ( 0 ) = sqrt { 0 } = 0) and (f ( 4 ) = sqrt { 4 } = 2). Recall the graph of the square root function.

The domain and range both consist of real numbers greater than or equal to zero: ([0, ∞)). To determine the domain of a function involving a square root we look at the radicand and find the values that produce nonnegative results.

Example (PageIndex{3}):

Determine the domain of the function defined by (f ( x ) = sqrt { 2 x + 3 }).

Solution

Here the radicand is (2x + 3). This expression must be zero or positive. In other words,

(2 x + 3 geq 0)

Solve for (x).

Answer:

Domain: (left[ - frac { 3 } { 2 } , infty ight))

A cube root5 of a number is a number that when multiplied by itself three times yields the original number. Furthermore, we denote a cube root using the symbol (sqrt [ 3 ] { }), where (3) is called the index6. For example,

(sqrt [ 3 ] { 64 } = 4 , ext { because } 4 ^ { 3 } = 64)

The product of three equal factors will be positive if the factor is positive and negative if the factor is negative. For this reason, any real number will have only one real cube root. Hence the technicalities associated with the principal root do not apply. For example,

In general, given any real number (a), we have the following property:

(sqrt [ 3 ] { a ^ { 3 } } = a quad ext { if } quad a in R)

When simplifying cube roots, look for factors that are perfect cubes.

Example (PageIndex{4}):

Evaluate.

  1. (sqrt [ 3 ] { 8 })
  2. (sqrt [ 3 ] { 0 })
  3. (sqrt [ 3 ] { frac { 1 } { 27 } })
  4. (sqrt [ 3 ] { - 1 })
  5. (sqrt [ 3 ] { - 125 })

Solution

  1. (sqrt [ 3 ] { 8 } = sqrt [ 3 ] { 2 ^ { 3 } } = 2)
  2. (sqrt [ 3 ] { 0 } = sqrt [ 3 ] { 0 ^ { 3 } } = 0)
  3. (sqrt [ 3 ] { frac { 1 } { 27 } } = sqrt [ 3 ] { left( frac { 1 } { 3 } ight) ^ { 3 } } = frac { 1 } { 3 })
  4. (sqrt [ 3 ] { - 1 } = sqrt [ 3 ] { ( - 1 ) ^ { 3 } } = - 1)
  5. (sqrt [ 3 ] { - 125 } = sqrt [ 3 ] { ( - 5 ) ^ { 3 } } = - 5)

It may be the case that the radicand is not a perfect square or cube. If an integer is not a perfect power of the index, then its root will be irrational. For example, (sqrt [ 3 ] { 2 }) is an irrational number that can be approximated on most calculators using the root button (sqrt [ x ] { }).Depending on the calculator, we typically type in the index prior to pushing the button and then the radicand as follows:

(3 quadsqrt [ x ] {y }quad2quad=)

Therefore, we have

(sqrt [ 3 ] { 2 } approx 1.260 , quad ext { because } quad 1.260 ^{wedge} 3 approx 2)

Since cube roots can be negative, zero, or positive we do not make use of any absolute values.

Example (PageIndex{5}):

Simplify: (sqrt [ 3 ] { ( y - 7 ) ^ { 3 } }).

Solution

The cube root of a quantity cubed is that quantity.

Answer:

(y-7)

Exercise (PageIndex{1})

Evaluate: (sqrt [ 3 ] { - 1000 }).

Answer

(=10)

www.youtube.com/v/B06NIs-3gig

Next, consider the cube root function7:

(f ( x ) = sqrt [ 3 ] { x } quadcolor{Cerulean}{Cube:root:function.})

Since the cube root could be either negative or positive, we conclude that the domain consists of all real numbers. Sketch the graph by plotting points. Choose some positive and negative values for (x), as well as zero, and then calculate the corresponding (y)-values.

(x)(f(x))(f ( x ) = sqrt [ 3 ] { x })(color{Cerulean}{Ordered:Pairs})
(-8)(color{Cerulean}{-2})((-8,-2))
(-1)(color{Cerulean}{-1})((-1,-1))
(0)(color{Cerulean}{0})(f ( 0 ) = sqrt [ 3 ] { 0 } = 0)((0,0))
(1)(color{Cerulean}{1})(f ( 1 ) = sqrt [ 3 ] { 1 } = 1)((1,1))
(8)(color{Cerulean}{2})(f ( 8 ) = sqrt [ 3 ] { 8 } = 2)((8,2))
Table (PageIndex{1})

Plot the points and sketch the graph of the cube root function.

The graph passes the vertical line test and is indeed a function. In addition, the range consists of all real numbers.

Example (PageIndex{6}):

Given (g ( x ) = sqrt [ 3 ] { x + 1 } + 2), find (g ( - 9 ) , g ( - 2 ) , g ( - 1 )), and (g(0)). Sketch the graph of (g).

Solution

Replace (x) with the given values.

(x)(g(x))(g ( x ) = sqrt [ 3 ] { x + 1 } + 2)(color{Cerulean}{Ordered:Pairs})
(-9)(color{Cerulean}{0})((-9,0))
(-2)(color{Cerulean}{1})((-2,1))
(-1)(color{Cerulean}{2})((-1,2))
(0)(color{Cerulean}{3})(g ( color{OliveGreen}{0}color{black}{ )} = sqrt [ 3 ] { color{OliveGreen}{0}color{black}{ +} 1 } + 2 = sqrt [ 3 ] { 1 } + 2 = 1 + 2 = 3)((0,3))
Table (PageIndex{2})

We can also sketch the graph using the following translations:

(egin{array} { l } { y = sqrt [ 3 ] { x } quadquadquadquad color{Cerulean} { Basic: cube :root: function } } { y = sqrt [ 3 ] { x + 1 } quad quad:color{Cerulean} { Horizontal: shift: left: 1: unit } } { y = sqrt [ 3 ] { x + 1 } + 2 :::color{Cerulean} { Vertical: shift: up: 2: units } } end{array})

Answer:

(n)th Roots

For any integer (n ≥ 2), we define an (n)th root8 of a positive real number as a number that when raised to the (n)th power yields the original number. Given any nonnegative real number (a), we have the following property:

(sqrt [ n ] { a ^ { n } } = a , quad ext { if } quad a geq 0)

Here n is called the index and (a^{n}) is called the radicand. Furthermore, we can refer to the entire expression (sqrt [ n ] { A }) as a radical9. When the index is an integer greater than or equal to (4), we say “fourth root,” “fifth root,” and so on. The (n)th root of any number is apparent if we can write the radicand with an exponent equal to the index.

Example (PageIndex{7}):

Simplify:

  1. (sqrt [ 4 ] { 81 })
  2. (sqrt [ 5 ] { 32 })
  3. (sqrt [ 7 ] { 1 })
  4. (sqrt [ 4 ] { frac { 1 } { 16 } })

Solution

  1. (sqrt [ 4 ] { 81 } = sqrt [ 4 ] { 3 ^ { 4 } } = 3)
  2. (sqrt [ 5 ] { 32 } = sqrt [ 5 ] { 2 ^ { 5 } } = 2)
  3. (sqrt [ 7 ] { 1 } = sqrt [ 7 ] { 1 ^ { 7 } } = 1)
  4. (sqrt [ 4 ] { frac { 1 } { 16 } } = sqrt [ 4 ] { left( frac { 1 } { 2 } ight) ^ { 4 } } = frac { 1 } { 2 })

Note

If the index is (n = 2), then the radical indicates a square root and it is customary to write the radical without the index; (sqrt [ 2 ] { a } = sqrt { a }).

We have already taken care to define the principal square root of a real number. At this point, we extend this idea to nth roots when n is even. For example, (3) is a fourth root of (81), because (3^{4} = 81). And since ((−3)^{4} = 81), we can say that (−3) is a fourth root of (81) as well. Hence we use the radical sign (sqrt [ n ] { }) to denote the principal (nonnegative) (n)th root10 when (n) is even. In this case, for any real number (a), we use the following property:

(sqrt [ n ] { a ^ { n } } = | a | quad color{Cerulean} { When:n:is:even } )

For example,

The negative (n)th root, when (n) is even, will be denoted using a negative sign in front of the radical (- sqrt [ n ] { }).

We have seen that the square root of a negative number is not real because any real number that is squared will result in a positive number. In fact, a similar problem arises for any even index:

We can see that a fourth root of (−81) is not a real number because the fourth power of any real number is always positive.

You are encouraged to try all of these on a calculator. What does it say?

Example (PageIndex{8}):

Simplify.

  1. (sqrt [ 4 ] { ( - 10 ) ^ { 4 } })
  2. (sqrt [ 4 ] { - 10 ^ { 4 } })
  3. (sqrt [ 6 ] { ( 2 y + 1 ) ^ { 6 } })

Solution

Since the indices are even, use absolute values to ensure nonnegative results.

  1. (sqrt [ 4 ] { ( - 10 ) ^ { 4 } } = | - 10 | = 10)
  2. (sqrt [ 4 ] { - 10 ^ { 4 } } = sqrt [ 4 ] { - 10,000 }) is not a real number.
  3. (sqrt [ 6 ] { ( 2 y + 1 ) ^ { 6 } } = | 2 y + 1 |)

When the index (n) is odd, the same problems do not occur. The product of an odd number of positive factors is positive and the product of an odd number of negative factors is negative. Hence when the index (n) is odd, there is only one real (n)th root for any real number (a). And we have the following property:

(sqrt [ n ] { a ^ { n } } = a quad color{Cerulean} { When : n:is:odd})

Example (PageIndex{9}):

Simplify.

  1. (sqrt [ 5 ] { ( - 10 ) ^ { 5 } })
  2. (sqrt [ 5 ] { - 32 })
  3. (sqrt [ 7 ] { ( 2 y + 1 ) ^ { 7 } })

Solution

Since the indices are odd, the absolute value is not used.

  1. (sqrt [ 5 ] { ( - 10 ) ^ { 5 } } = - 10)
  2. (sqrt [ 5 ] { - 32 } = sqrt [ 5 ] { ( - 2 ) ^ { 5 } } = - 2)
  3. (sqrt [ 7 ] { ( 2 y + 1 ) ^ { 7 } } = 2 y + 1)

In summary, for any real number (a) we have,

(egin{aligned} sqrt [ n ] { a^ { n } } & = | a | color{Cerulean}::: { When : n: is: even } sqrt [ n ] {a^ { n } } & = a quad: color{Cerulean} { When : n: is: odd } end{aligned})

When (n) is odd, the (n)th root is positive or negative depending on the sign of the radicand.

When (n) is even, the (n)th root is positive or not real depending on the sign of the radicand.

Exercise (PageIndex{2})

Simplify: (- 8 sqrt [ 5 ] { - 32 }).

Answer

(16)

www.youtube.com/v/Ik1xXgq18f0

Simplifying Radicals

It will not always be the case that the radicand is a perfect power of the given index. If it is not, then we use the product rule for radicals11 and the quotient rule for radicals12 to simplify them. Given real numbers (sqrt [ n ] { A }) and (sqrt [ n ] { B }),

Product Rule for Radicals:(sqrt [ n ] { A cdot B } = sqrt [ n ] { A } cdot sqrt [ n ] { B })
Quotient Rule for Radicals:(sqrt [ n ] { frac { A } { B } } = frac { sqrt [ n ] { A } } { sqrt [ n ] { B } })
Table (PageIndex{3})

A radical is simplified13 if it does not contain any factors that can be written as perfect powers of the index.

Example (PageIndex{10}):

Simplify: (sqrt { 150 }).

Solution

Here (150) can be written as (2 cdot 3 cdot 5 ^ { 2 }).

(egin{aligned} sqrt { 150 } & = sqrt { 2 cdot 3 cdot 5 ^ { 2 } }quadquad color{Cerulean} { Apply: the: product :rule: for: radicals.} & = sqrt { 2 cdot 3 } cdot sqrt { 5 ^ { 2 } }quad: color{Cerulean} { Simplify. } & = sqrt { 6 } cdot 5 & = 5 sqrt { 6 } end{aligned})

We can verify our answer on a calculator:

(sqrt { 150 } approx 12.25 quad ext { and }quad 5 sqrt { 6 } approx 12.25)

Also, it is worth noting that

(12.25 ^ { 2 } approx 150)

Answer:

(5 sqrt { 6 })

Note

(5 sqrt { 6 }) is the exact answer and (12.25) is an approximate answer. We present exact answers unless told otherwise.

Example (PageIndex{11}):

Simplify: (sqrt [ 3 ] { 160 }).

Solution

Use the prime factorization of (160) to find the largest perfect cube factor:

(egin{aligned} 160 & = 2 ^ { 5 } cdot 5 & = color{Cerulean}{2 ^ { 3} }color{black}{ cdot} 2 ^ { 2 } cdot 5 end{aligned})

Replace the radicand with this factorization and then apply the product rule for radicals.

(egin{aligned} sqrt [ 3 ] { 160 } & = sqrt [ 3 ] { 2 ^ { 3 } cdot 2 ^ { 2 } cdot 5 } quadquadcolor{Cerulean} { Apply:the: product: rule: for: radicals.} & = sqrt [ 3 ] { 2 ^ { 3 } } cdot sqrt [ 3 ] { 2 ^ { 2 } cdot 5 }quad color{Cerulean} { Simplify. } & = 2 cdot sqrt [ 3 ] { 20 } end{aligned})

We can verify our answer on a calculator.

(sqrt [ 3 ] { 160 } approx 5.43 ext { and } 2 sqrt [ 3 ] { 20 } approx 5.43)

Answer:

(2 sqrt [ 3 ] { 20 })

Example (PageIndex{12}):

Simplify: (sqrt [ 5 ] { - 320 }).

Solution

Here we note that the index is odd and the radicand is negative; hence the result will be negative. We can factor the radicand as follows:

Then simplify:

Answer:

(- 2 sqrt [ 5 ] { 10 })

Example (PageIndex{13}):

Simplify: (sqrt [ 3 ] { - frac { 8 } { 64 } }).

Solution

In this case, consider the equivalent fraction with (−8 = (−2)^{3}) in the numerator and (64 = 4^{3}) in the denominator and then simplify.

Answer:

(-frac{1}{2})

Exercise (PageIndex{3})

Simplify: (sqrt [ 4 ] { frac { 80 } { 81 } })

Answer

(frac { 2 sqrt [ 4 ] { 5 } } { 3 })

www.youtube.com/v/8CwbDBFO2FQ

Key Takeaways

  • To simplify a square root, look for the largest perfect square factor of the radicand and then apply the product or quotient rule for radicals.
  • To simplify a cube root, look for the largest perfect cube factor of the radicand and then apply the product or quotient rule for radicals.
  • When working with nth roots, (n) determines the definition that applies. We use (sqrt [ n ] { a ^ { n } } = a _ { 1 }) when (n) is odd and (sqrt [ n ] { a ^ { n } } = | a | ) when (n) is even.
  • To simplify (n)th roots, look for the factors that have a power that is equal to the index (n) and then apply the product or quotient rule for radicals. Typically, the process is streamlined if you work with the prime factorization of the radicand.

Exercise (PageIndex{4})

Simplify.

  1. (sqrt { 36 })
  2. (sqrt { 100 })
  3. (sqrt { frac { 4 } { 9 } })
  4. (sqrt { frac { 1 } { 64 } })
  5. (- sqrt { 16 })
  6. (- sqrt { 1 })
  7. (sqrt { ( - 5 ) ^ { 2 } })
  8. (sqrt { ( - 1 ) ^ { 2 } })
  9. (sqrt { - 4 })
  10. (sqrt { - 5 ^ { 2 } })
  11. (- sqrt { ( - 3 ) ^ { 2 } })
  12. (- sqrt { ( - 4 ) ^ { 2 } })
  13. (sqrt { x ^ { 2 } })
  14. (sqrt { ( - x ) ^ { 2 } })
  15. (sqrt { ( x - 5 ) ^ { 2 } })
  16. (sqrt { ( 2 x - 1 ) ^ { 2 } })
  17. (sqrt [ 3 ] { 64 })
  18. (sqrt [ 3 ] { 216 })
  19. (sqrt [ 3 ] { - 216 })
  20. (sqrt [ 3 ] { - 64 })
  21. (sqrt [ 3 ] { - 8 })
  22. (sqrt [ 3 ] { 1 })
  23. (- sqrt [ 3 ] { ( - 2 ) ^ { 3 } })
  24. (- sqrt [ 3 ] { ( - 7 ) ^ { 3 } })
  25. (sqrt [ 3 ] { frac { 1 } { 8 } })
  26. (sqrt [ 3 ] { frac { 8 } { 27 } })
  27. (sqrt [ 3 ] { ( - y ) ^ { 3 } })
  28. (- sqrt [ 3 ] { y ^ { 3 } })
  29. (sqrt [ 3 ] { ( y - 8 ) ^ { 3 } })
  30. (sqrt [ 3 ] { ( 2 x - 3 ) ^ { 3 } })
Answer

1. (6)

3. (frac{2}{3})

5. (−4)

7. (5)

9. Not a real number

11. (−3)

13. (|x|)

15. (|x − 5|)

17. (4)

19. (−6)

21. (−2)

23. (2)

25. (frac{1}{2})

27. (−y)

29. (y − 8)

Exercise (PageIndex{5})

Determine the domain of the given function.

  1. (g ( x ) = sqrt { x + 5 })
  2. (g ( x ) = sqrt { x - 2 })
  3. (f ( x ) = sqrt { 5 x + 1 })
  4. (f ( x ) = sqrt { 3 x + 4 })
  5. (g ( x ) = sqrt { - x + 1 })
  6. (g ( x ) = sqrt { - x - 3 })
  7. (h ( x ) = sqrt { 5 - x })
  8. (h ( x ) = sqrt { 2 - 3 x })
  9. (g ( x ) = sqrt [ 3 ] { x + 4 })
  10. (g ( x ) = sqrt [ 3 ] { x - 3 })
Answer

1. ([ - 5 , infty ))

3. (left[ - frac { 1 } { 5 } , infty ight))

5. (( - infty , 1 ])

7. (( - infty , 5 ])

9. (( - infty , infty ))

Exercise (PageIndex{6})

Evaluate given the function definition.

  1. Given (f ( x ) = sqrt { x - 1 }), find (f ( 1 ) , f ( 2 )), and (f ( 5 ))
  2. Given (f ( x ) = sqrt { x + 5 }), find (f ( - 5 ) , f ( - 1 )), and (f ( 20 ))
  3. Given (f ( x ) = sqrt { x } + 3), find (f ( 0 ) , f ( 1 )), and (f(16))
  4. Given (f ( x ) = sqrt { x } - 5), find (f ( 0 ) , f ( 1 )), and (f(25))
  5. Given (g ( x ) = sqrt [ 3 ] { x }), find (g ( - 1 ) , g ( 0 )), and (g(1))
  6. Given (g ( x ) = sqrt [ 3 ] { x } - 2) find (g ( - 1 ) , g ( 0 )), and (g(8))
  7. Given (g ( x ) = sqrt [ 3 ] { x + 7 }), find (g ( - 15 ) , g ( - 7 )), and (g(20))
  8. Given (g ( x ) = sqrt [ 3 ] { x - 1 } + 2), find (g ( 0 ) , g ( 2 ) ), and (g(9))
Answer

1. (f ( 1 ) = 0 ; f ( 2 ) = 1 ; f ( 5 ) = 2)

3. (f ( 0 ) = 3 ; f ( 1 ) = 4 ; f ( 16 ) = 7)

5. (g ( - 1 ) = - 1 ; g ( 0 ) = 0 ; g ( 1 ) = 1)

7. (g ( - 15 ) = - 2 ; g ( - 7 ) = 0 ; g ( 20 ) = 3)

Exercise (PageIndex{7})

Sketch the graph of the given function and give its domain and range.

  1. (f ( x ) = sqrt { x + 9 })
  2. (f ( x ) = sqrt { x - 3 })
  3. (f ( x ) = sqrt { x - 1 } + 2)
  4. (f ( x ) = sqrt { x + 1 } + 3)
  5. (g ( x ) = sqrt [ 3 ] { x - 1 })
  6. (g ( x ) = sqrt [ 3 ] { x + 1 })
  7. (g ( x ) = sqrt [ 3 ] { x } - 4)
  8. (g ( x ) = sqrt [ 3 ] { x } + 5)
  9. (g ( x ) = sqrt [ 3 ] { x + 2 } - 1)
  10. (g ( x ) = sqrt [ 3 ] { x - 2 } + 3)
  11. (f ( x ) = - sqrt [ 3 ] { x })
  12. (f ( x ) = - sqrt [ 3 ] { x - 1 })
Answer

1. Domain: ([ - 9 , infty )); range: ([ 0 , infty ))

3. Domain: ([ 1 , infty )); range: ([ 2 , infty ))

5. Domain: (mathbb { R }); range; (mathbb { R })

7. Domain: (mathbb { R }); range; (mathbb { R })

9. Domain: (mathbb { R }); range; (mathbb { R })

11. Domain: (mathbb { R }); range; (mathbb { R })

Exercise (PageIndex{8})

Simplify.

  1. (sqrt [ 4 ] { 64 })
  2. (sqrt [ 4 ] { 16 })
  3. (sqrt [ 4 ] { 625 })
  4. (sqrt [ 4 ] { 1 })
  5. (sqrt [ 4 ] { 256 })
  6. (sqrt [ 4 ] { 10,000 })
  7. (sqrt [ 5 ] { 243 })
  8. (sqrt [ 5 ] { 100,000 })
  9. (sqrt [ 5 ] { frac { 1 } { 32 } })
  10. (sqrt [ 5 ] { frac { 1 } { 243 } })
  11. (- sqrt [ 4 ] { 16 })
  12. (- sqrt [ 6 ] { 1 })
  13. (sqrt [ 5 ] { - 32 })
  14. (sqrt [ 5 ] { - 1 })
  15. (sqrt { - 1 })
  16. (sqrt [ 4 ] { - 16 })
  17. (- 6 sqrt [ 3 ] { - 27 })
  18. (- 5 sqrt [ 3 ] { - 8 })
  19. (2 sqrt [ 3 ] { - 1,000 })
  20. (7 sqrt [ 5 ] { - 243 })
  21. (6 sqrt [ 4 ] { - 16 })
  22. (12 sqrt [ 6 ] { - 64 })
  23. (sqrt [ 3 ] { frac { 25 } { 16 } })
  24. (6 sqrt { frac { 16 } { 9 } })
  25. (5 sqrt [ 3 ] { frac { 27 } { 125 } })
  26. (7 sqrt [ 5 ] { frac { 32 } { 7 ^ { 5 } } })
  27. (- 5 sqrt [ 3 ] { frac { 8 } { 27 } })
  28. (- 8 sqrt [ 4 ] { frac { 625 } { 16 } })
  29. (2 sqrt [ 5 ] { 100,000 })
  30. (2 sqrt [ 7 ] { 128 })
Answer

1. (4)

3. (5)

5. (4)

7. (3)

9. (frac{1}{2})

11. (−2)

13. (−2)

15. Not a real number

17. (18)

19. (−20)

21. Not a real number

23. (frac{15}{4})

25. (3)

27. (−frac{10}{3})

29. (20)

Exercise (PageIndex{9})

Simplify.

  1. (sqrt { 96 })
  2. (sqrt { 500 })
  3. (sqrt { 480 })
  4. (sqrt { 450 })
  5. (sqrt { 320 })
  6. (sqrt { 216 })
  7. (5 sqrt { 112 })
  8. (10 sqrt { 135 })
  9. (- 2 sqrt { 240 })
  10. (- 3 sqrt { 162 })
  11. (sqrt { frac { 150 } { 49 } })
  12. (sqrt { frac { 200 } { 9 } })
  13. (sqrt { frac { 675 } { 121 } })
  14. (sqrt { frac { 192 } { 81 } })
  15. (sqrt [ 3 ] { 54 })
  16. (sqrt [ 3 ] { 24 })
  17. (sqrt [ 3 ] { 48 })
  18. (sqrt [ 3 ] { 81 })
  19. (sqrt [ 3 ] { 40 })
  20. (sqrt [ 3 ] { 120 })
  21. (sqrt [ 3 ] { 162 })
  22. (sqrt [ 3 ] { 500 })
  23. (sqrt [ 3 ] { frac { 54 } { 125 } })
  24. (sqrt [ 3 ] { frac { 40 } { 343 } })
  25. (5 sqrt [ 3 ] { - 48 })
  26. (2 sqrt [ 3 ] { - 108 })
  27. (8 sqrt [ 4 ] { 96 })
  28. (7 sqrt [ 4 ] { 162 })
  29. (sqrt [ 5 ] { 160 })
  30. (sqrt [ 5 ] { 486 })
  31. (sqrt [ 5 ] { frac { 224 } { 243 } })
  32. (sqrt [ 5 ] { frac { 5 } { 32 } })
  33. (sqrt [ 5 ] { - frac { 1 } { 32 } })
  34. (sqrt [ 6 ] { - frac { 1 } { 64 } })
Answer

1. (4 sqrt { 6 })

3. (4 sqrt { 30 })

5. (8 sqrt { 5 })

7. (20 sqrt { 7 })

9. (- 8 sqrt { 15 })

11. (frac { 5 sqrt { 6 } } { 7 })

13. (frac { 15 sqrt { 3 } } { 11 })

15. (3 sqrt [ 3 ] { 2 })

17. (2 sqrt [ 3 ] { 6 })

19. (2 sqrt [ 3 ] { 5 })

21. (3 sqrt [ 3 ] { 6 })

23. (frac { 3 sqrt [ 3 ] { 2 } } { 5 })

25. (- 10 sqrt [ 3 ] { 6 })

27. (16 sqrt [ 4 ] { 6 })

29. (2 sqrt [ 5 ] { 5 })

31. (frac { 2 sqrt [ 5 ] { 7 } } { 3 })

33. (- frac { 1 } { 2 })

Exercise (PageIndex{10})

Simplify. Give the exact answer and the approximate answer rounded to the nearest hundredth.

  1. (sqrt { 60 })
  2. (sqrt { 600 })
  3. (sqrt { frac { 96 } { 49 } })
  4. (sqrt { frac { 192 } { 25 } })
  5. (sqrt [ 3 ] { 240 })
  6. (sqrt [ 3 ] { 320 })
  7. (sqrt [ 3 ] { frac { 288 } { 125 } })
  8. (sqrt [ 3 ] { frac { 625 } { 8 } })
  9. (sqrt [ 4 ] { 486 })
  10. (sqrt [ 5 ] { 288 })
Answer

1. (2 sqrt { 15 } ; 7.75)

3. (frac { 4 sqrt { 6 } } { 7 } ; 1.40)

5. (2 sqrt [ 3 ] { 30 } ; 6.21)

7. (frac { 2 sqrt [ 3 ] { 36 } } { 5 } ; 1.32)

9. (3 sqrt [ 4 ] { 6 } ; 4.70)

Exercise (PageIndex{11})

Rewrite the following as a radical expression with coeffecient (1).

  1. (2 sqrt { 15 })
  2. (3 sqrt { 7 })
  3. (5 sqrt { 10 })
  4. (10 sqrt { 3 })
  5. (2 sqrt [ 3 ] { 7 })
  6. (3 sqrt [ 3 ] { 6 })
  7. (2 sqrt [ 4 ] { 5 })
  8. (3sqrt [ 4 ] { 2 })
  9. Each side of a square has a length that is equal to the square root of the square’s area. If the area of a square is (72) square units, find the length of each of its sides.
  10. Each edge of a cube has a length that is equal to the cube root of the cube’s volume. If the volume of a cube is (375) cubic units, find the length of each of its edges.
  11. The current (I) measured in amperes is given by the formula (I = sqrt { frac { P } { R } }) where (P) is the power usage measured in watts and (R) is the resistance measured in ohms. If a (100) watt light bulb has (160) ohms of resistance, find the current needed. (Round to the nearest hundredth of an ampere.)
  12. The time in seconds an object is in free fall is given by the formula (t = frac { sqrt { s } } { 4 }) where (s) represents the distance in feet the object has fallen. How long will it take an object to fall to the ground from the top of an (8)-foot stepladder? (Round to the nearest tenth of a second.)
Answer

1. (sqrt { 60 })

3. (sqrt { 250 })

5. (sqrt [ 3 ] { 56 })

7. (sqrt [ 4 ] { 80 })

9. (6 sqrt { 2 }) units

11. (0.79) ampere

Exercise (PageIndex{12})

  1. Explain why there are two real square roots for any positive real number and one real cube root for any real number.
  2. What is the square root of (1) and what is the cube root of (1)? Explain why.
  3. Explain why (sqrt { - 1 }) is not a real number and why (sqrt [ 3 ] { - 1 }) is a real number.
  4. Research and discuss the methods used for calculating square roots before the common use of electronic calculators.
Answer

1. Answer may vary

3. Answer may vary

Footnotes

1A number that when multiplied by itself yields the original number.

2The positive square root of a positive real number, denoted with the symbol (√).

3The expression (A) within a radical sign, (sqrt [ n ] { A }).

4The function defined by (f ( x ) = sqrt { x }).

5A number that when used as a factor with itself three times yields the original number, denoted with the symbol (sqrt [ 3 ] { }).

6The positive integer (n) in the notation (sqrt [ n ] { }) that is used to indicate an nth root.

7The function defined by (f ( x ) = sqrt [ 3 ] { x }).

8A number that when raised to the (n)th power ((n ≥ 2)) yields the original number.

9Used when referring to an expression of the form (sqrt [ n ] { A }).

10The positive (n)th root when (n) is even.

11Given real numbers (sqrt [ n ] { A }) and (sqrt [ n ] { B }),(sqrt [ n ] { A cdot B } = sqrt [ n ] { A } cdot sqrt [ n ] { B }).

12Given real numbers (sqrt [ n ] { A }) and (sqrt [ n ] { B }),(sqrt [ n ] { frac { A } { B } } = frac { sqrt [ n ] { A } } { sqrt [ n ] { B } }) where (B ≠ 0).

13A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.


Radicals and Absolute Values - Concept

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Since any even-numbered root must be a positive number (otherwise it is imaginary), absolute value must be used when simplifying roots with variables, which ensures the answer is positive. When working with radical expressions this requirement does not apply to any odd root because odd roots exist for negative numbers. Additionally, absolute value is not needed if an even number of a variable come out of the root - the answer must be positive.

The absolute value and square roots, so what we're going to do now is talk about absolute values and sometimes what happens is we actually need an absolute value when we give our answer and so what we're going to do is look at a number of examples and talk about when we need them and when we don't.
Okay so it's starting off, square root of 4, easy example we know the square root of 4 is 2 because 2 times 2 is equal to 4. Okay next, square root of -4, we need 2 numbers that will give us -4 that's not going to happen okay this is a not a real number okay later on we'll actually talk about how we can do this but we're not there yet okay? So square root of 3 squared. 3 squared is 9, square root of nine is 3. Square root of negative 3 squared. When we square a negative number, we actually get a positive so -3 times -3 is 9 square root of 9 is again 3. Okay, cube root of 8 has three 2's in eight so this comes out to be 2 and the cube root of negative 8 is -2. Negative 2 times itself 3 times is negative eight so what we have looked at here is when we have a odd root okay cube root here we can have a positive answer or a negative answer.
Okay when we have a even root all of our answers have to be positive okay? So these are numeric representations but now we're going to variables. Okay, so let's go over here the square root of x squared okay that is made up of 2 x's so we know that we can simplify this as an x. The problem is is that we don't know if x is positive or negative right? Say x was negative 3, just like we had over here. What happens is we are squaring it so that will become positive and then taking the square root of it so it's actually going to stay positive because we don't know if x is positive or negative, we have to put absolute value signs from the outside to make that term positive okay, so square root of x cubed. We can take out one x and we're still left with one x on the inside but we know that this has to be positive because we can't take a negative out of the square root so once again we have to put in absolute value signs okay because wherever comes out of the square root has to be positive. Okay, what about square root of x to the fourth. Okay we know this is 4 xs so we can take out 2 of them leaving us with x squared. Do we need absolute values in this case? No because x squared is always going to be become positive okay? So whenever we need absolute values is basically a one sort of rule of thumb that I always use is whenever you're taking an even root, so here this is the square root there's actually a little invisible 2 out here and whenever taking a even root and we have an add power on a variable, okay so here we have a single x, single x, x squared. If we had to take out an x cubed we would need an absolute value. If we take out an x the fourth we won't because the fourth will always be positive.
We actually never need a absolute value when we're dealing with an odd root and we'll do one example more example say cube root of x to the third. x to the third is 3 x's so this actually equal to x but because we're dealing with an odd root the cube root we do not need an absolute value because we can actually get positive or negative numbers to come out of this okay? So basically whenever you are dealing with the absolute sorry the square root of a variable, if you have an even root and you get out and odd power you always include absolute values.


Simplifying Radicals

Rationalize the Denominator

roots of the number (or expression) in the radicand.

For example , the cube root of 8 stands for the number that can be multiplied by itself three times (cubed) to equal 8:

Since, 2 * 2 * 2 = 8, the cube root of 8 is 2.

The 3 in the expression is called the root index, and the 8 is called the radicand.

What does "Rationalizing the Denominator" mean?

Rationalizing the Denominator simply means to remove all radicals from the denominator of a fraction without changing the value of the fraction.

When the denominator is rationalized, the original fraction is converted to the simplest equivalent fraction which does not have radicals in the denominator.

By removing all radicals from the denominator, all numbers in the denominator will be converted rational numbers (hence the term, "Rationalizing the Denominator").

(Note that the square root of 2 is an
irrational number - a non-terminating
decimal without a repeating pattern.)

(Note that the denominator is now the
rational number 2.)

Why Rationalize the Denominator of a fraction?

Why not leave the denominator alone?

What difference does it make?

What is the point of Rationalizing the Denominator?

Is it ever necessary to rationalize the numerator of a fraction instead of a denominator?

How to rationalize the denominator of a fraction


Video demonstrating How to Rationalize the Denominator - click here

Rationalize the denominator of:

Rationalize the denominator of:

Rationalize the denominator of:

The final answer is:

Rationalize the denominator of:

Rationalize the denominator of:

Rationalize the denominator of:

The final answer is:


Table of Contents

Chapter 1 Order Of Operations

1.1.1 Addition and Multiplication Which is correct?

1.1.2 Addition and Exponents Which is correct?

1.1.3 Multiply and Divide from left to right Which is correct?

1.1.4 Multiplication and Exponents Which is correct?

Chapter 2 Number Properties

2.1.1 Multiplying fractions Which is better?

2.2.1 Division vs. multiplication by the reciprocal How do they differ?

2.2.2 Multiplication vs. division by the reciprocal (with integers) How do they differ?

2.2.3 Multiplication vs. division by the reciprocal (with fractions) How do they differ?

2.2.4 Simplifying fractional expressions Which is correct?

2.3.1 Addition – Add 3 numbers in any order Why does it work?

2.3.2 Addition – Addition of negative numbers Why does it work?

2.3.3 Subtraction – Subtraction is not associative Which is correct?

2.3.4 Multiplication – Multiply 3 numbers in any order Why does it work?

2.3.5 Division – Division is not associative Which is correct?

2.4.1 Addition of Negatives – If rewrite subtraction as addition, can use commutative property Which is correct?

2.4.2 Subtraction – Subtraction is not commutative How do they differ?

2.5.1 Distributive Property and Order of Operations – Distribute vs. simplify in parentheses first Why does it work?

2.5.2 Common error: Fail to distribute over both terms Which is correct?

2.5.3 Common error: Incorrectly using the distributive property with multiplication Which is correct?

2.6.1 Simplifying expressions with absolute value Which is correct?

Operations with Negative Numbers

2.7.1 Subtracting a negative vs. rewriting subtraction as addition of the opposite Which is correct?

Chapter 3 Linear Equations

Solving Multi-Step Equations

3.1.1 Subtraction – Subtract first vs. distribute first Which is better?

3.1.2 Fractions – Eliminate fractions first vs. find common denominator first Which is better?

Solving Multi-Step Equations

3.1.3 Incorrectly performing an operation on both sides of an equation Which is correct?

3.1.4 Fractions – Subtract first vs. find a common denominator first Which is better?

3.1.5 Division – Cross-multiply vs. multiply both sides of the equation by the same value Why does it work?

3.1.6 Subtract first vs. divide first (for equations of the form a(x + b) = c, where a is divisible by c) Which is better?

3.1.7 Distribute first vs. multiply first (for equations of the form a(x + b) = c, where a is a fraction) Which is better?

Solving Equations with Variables on Both Sides

3.2.1 Subtract first vs. distribute first Which is better?

3.2.2 Combining like terms Which is correct?

3.2.3 Performing same operation twice on one side Which is correct?

3.2.4 Combining terms with common factors Which is better?

Solving Literal Equations

3.3.1 Divide first vs. distribute first Which is better?

3.3.2 Move variables one at a time vs. all at once Which is better?

3.3.3 Subtract first vs. divide first Why does it work?

3.4.1 Multiply first vs. cross-multiply Why does it work?

3.4.2 Equivalent fractions Which is better?

3.4.3 Unit rate Which is better?

3.4.4 Multiplying fractions vs. solving a proportion Which is correct?

Chapter 4 Graphing Linear Equations And Introduction To Functions And Relations

4.1.1 Table of values vs. slope-intercept form Which is better?

4.1.2 Choosing x-values for the table Which is better?

4.2.1 Find a slope by graphing vs. by using the slope formula Why does it work?

Graphing Lines using Intercepts

4.3.1 Shortcut to finding the intercepts Why does it work?

4.4.1 Choose either point Why does it work?

4.4.2 Graph an equation given in point-slope form using the point-slope or the slope-intercept method Which is better?

Using Slope-Intercept Form

4.5.1 Given 2 points, find the y-intercept Which is better?

4.5.2 Forget which term represents the y-intercept Which is correct?

4.5.3 Confuse rise and run Which is correct?

4.5.4 Comparing the equations of lines with different slope values How do they differ?

4.5.5 Comparing the equations of lines with positive and negative slopes How do they differ?

4.5.6 Comparing the equations of lines with different y-intercepts How do they differ?

4.5.7 Comparing x-intercepts and y-intercepts How do they differ?

4.6.1 Given an equation in standard form, graph using slope-intercept vs. intercepts Which is better?

Slope of a Horizontal Line

4.7.1 Find slope by formula vs. t-table Why does it work?

4.8.1 Find slope by formula vs. t-table Why does it work?

4.9.1 Determine whether a relation is a function How do they differ?

4.9.2 Domain vs. range How do they differ?

Solving Inequalities using Multiplication and Division

5.1.1 Division by a positive vs. a negative number Which is correct?

5.1.2 Why we can ‘flip’ the inequality sign in division Why does it work?

5.1.3 ‘Flipping’ the inequality sign when dividing by a positive value Which is correct?

Absolute Value Inequalities

5.2.1 Definition Why does it work?

5.2.2 Greater than Why does it work?

5.2.3 Addition Why does it work?

5.3.1 Error shading by sign Which is correct?

5.3.2 Graphing an equation vs. graphing an inequality How do they differ?

5.3.3 Graphing inequalities with “and” vs. “or” How do they differ?

Chapter 6 Systems Of Equations

6.1.1 Substitute the first value you get into either equation to get the second value Why does it work?

6.1.2 Solve for x vs. solve for y Why does it work?

6.1.3 Substitute for a more or less efficient variable Which is better?

6.2.1 Substitute the first value you get into either equation to get the second value Why does it work?

6.2.2 Eliminate x vs. eliminate y How do they differ?

6.2.3 Common error Which is correct?

6.2.4 Why elimination works Why does it work?

Substitution vs. Elimination

6.3.1 Both methods work Which is better?

6.3.2 Identifying when substitution is preferable Which is better?

6.3.3 Identifying when elimination is preferable Which is better?

Adding and Subtracting Polynomials

7.1.1 Addition – Missing terms Which is better?

7.1.2 Subtraction – Missing terms Which is better?

Multiplying and Dividing Polynomials

7.2.1 Multiplying Polynomials – Area model vs. distributive property Why does it work?

7.2.2 Multiplying Polynomials – Distributive property vs. FOIL Why does it work?

7.2.3 Multiplying Polynomials – Forgetting to distribute to each term Which is correct?

7.3.1 Dividing Polynomials – Common error (no placeholders) Which is correct?

7.4.1 Finding GCF by factor tree vs. product pairs Which is better?

7.4.2 GCF of terms with variables by factor tree vs. product pairs Which is better?

7.4.3 Adding terms with common factors Which is better?

7.4.4 Adding terms with common factors Why does it work?

7.5.1 Factor a trinomial in two variables How do they differ?

7.5.2 Factor a trinomial with lead coefficient not 1: Factor by trial and error vs. factor first Which is better?

7.5.3 Factor a trinomial with lead coefficient not 1: Factor by trial and error vs. splitting the middle term Which is better?

7.5.4 Solve a quadratic equation with lead coefficient not 1: Factor out a common factor first vs. don’t Which is better?

7.5.5 Factor a trinomial with lead coefficient not 1:Factor by splitting the middle term first vs. by factoring out a common factor first Which is better?

8.1.1 Derivation of the quadratic formula by completing the square Why does it work?

8.2.1 Factoring vs. quadratic formula Which is better?

8.3.1 Why not to divide by a variable Which is correct?

8.4.1 Confuse sign of a Which is correct?

8.4.2 Confuse sign of b Which is correct?

8.4.3 Confuse sign of c Which is correct?

Graphing Quadratic Equations

8.5.1 Graphing quadratic equations with positive vs. negative coefficients for x 2 How do they differ?

8.5.2 Graphing quadratic equations with fractional vs. whole-number coefficients for x 2 How do they differ?

Graphing Quadratic Equations

8.5.3 How changing the coefficient of x 2 affects the graph of a quadratic equation How do they differ?

8.5.4 How adding a constant to the quadratic equation affects the graph How do they differ?

8.5.5 How adding vs. subtracting a constant to the quadratic equation affects the graph How do they differ?

8.5.6 How adding a constant to x affects the graph How do they differ?

8.5.7 How adding vs. subtracting a constant to x affects the graph How do they differ?

Operations with Terms with Exponents

9.1.1 Adding like terms with exponents Which is correct?

Multiplication Properties: Product of Powers Property

9.2.1 Multiplication with exponents Which is correct?

Multiplication Properties: Power of a Power Property

9.3.1 Rewriting terms with the same base Which is better?

9.3.2 Rewriting terms using perfect squares Which is better?

Multiplication Properties: Power of a Product Property

9.4.1 Power of a product Why does it work?

9.4.2 Apply the exponent to each number being multiplied Which is correct?

9.4.3 Order of operations vs. power of a product property Why does it work?

9.4.4 Simplifying expressions with variables Which is correct?

9.4.5 Simplifying expressions with addition of integers Which is correct?

9.4.6 Simplifying expressions involving products and exponents according to the order of operations Which is correct?

Division Properties: Quotient of Powers Property

9.5.1 Division with exponents Why does it work?

9.5.2 Division with exponents Which is correct?

Graphing Exponential Equations

9.6.1 Graphing linear vs. exponential functions How do they differ?

9.6.2 Graphing quadratic vs. exponential functions How do they differ?

Chapter 10 Radical Expressions and Equations

Properties of Radicals: Square Roots

10.1.1 Prime factor vs. perfect square Which is better?

10.1.2 Confusing the square root with division by 2 Which is correct?

10.1.3 Forgetting to simplify the square root of a*a as a Which is correct?

10.1.4 Estimating the value of a square root Why does it work?

10.1.5 Values for which the square root of a number is greater and less than the number How do they differ?

10.1.6 Square roots with numbers and variables Why does it work?

Product Property of Radicals

10.2.1 Multiplication of radicals Which is better?

10.2.2 Square root of a product vs. product of the square roots Why does it work?

Quotient Property of Radicals

10.3.1 Square root of a quotient vs. quotient of the square roots Why does it work?

10.3.2 Square root in the denominator of a fraction Which is better?

Addition and Subtraction of Radicals

10.4.1 Square root of a sum vs. sum of square roots Which is correct?

10.4.2 Combining two unlike radical terms Which is correct?

10.4.3 Combining three like radical terms Which is correct?

10.4.4 Factoring the radicand to combine like radical terms Which is correct?

10.4.5 Square root of a difference vs. difference of square roots Which is correct?

Multiplication and Division of Radicals

10.5.1 Simplifying expressions with sums vs. products in the radicand How do they differ?

Simplifying Radical Expressions

10.6.1 Factoring out radical terms as common factors Which is better?

10.6.2 Advantages of breaking terms into perfect squares first Which is better?

10.7.1 Distance formula and Pythagorean theorem Why does it work?

Chapter 11 Rational Expressions and Equations

11.1.1 Incorrect common denominator Which is correct?

11.1.2 Simplifying the fraction first vs. finding a common denominator first Which is better?

Multiplication and Division

11.2.1 Error simplifying the numerator and denominator Which is correct?


MathHelp.com

As you can see, simplifying radicals that contain variables works exactly the same way as simplifying radicals that contain only numbers. We factor, find things that are squares (or, which is the same thing, find factors that occur in pairs), and then we pull out one copy of whatever was squared (or of whatever we'd found a pair of).

Simplify

Looking at the numerical portion of the radicand, I see that the 12 is the product of 3 and 4 , so I have a pair of 2 's (so I can take a 2 out front) but a 3 left over (which will remain behind inside the radical).

Looking at the variable portion, I have two pairs of a 's I have three pairs of b 's, with one b left over and I have one pair of c 's, with one c left over. So the root simplifies as:

You are used to putting the numbers first in an algebraic expression, followed by any variables. But for radical expressions, any variables outside the radical should go in front of the radical, as shown above. Always put everything you take out of the radical in front of that radical (if anything is left inside it).

Simplify

Writing out the complete factorization would be a bore, so I'll just use what I know about powers. The 20 factors as 4 × 5 , with the 4 being a perfect square. The r 18 has nine pairs of r' s the s is unpaired and the t 21 has ten pairs of t 's, with one t left over. Then:

Technical point: Your textbook may tell you to "assume all variables are positive" when you simplify. Why? Because the square root of the square of a negative number is not the original number.

For instance, you could start with &ndash2 , square it to get +4 , and then take the square root of +4 (which is defined to be the positive root) to get +2 . You plugged in a negative and ended up with a positive.

We're applying a process that results in our getting the same numerical value, but it's always positive (or at least non-negative). Sound familiar? It should: it's how the absolute value works: |&ndash2| = +2 . Taking the square root of the square is in fact the technical definition of the absolute value.

But this technicality can cause difficulties if you're working with values of unknown sign that is, with variables. The |&ndash2| is +2 , but what is the sign on | x | ? You can't know, because you don't know the sign of x itself &mdash unless they specify that you should "assume all variables are positive", or at least non-negative (which means "positive or zero").


Multiplying Radical Expressions

1)

2)

On the outside, we only have a 5, so that stays the same (or you can think of it as 5.1). On the inside, 3(6)=18.can be simplified to Since the 5 and 3 are both on the outside, we finish by multiplying them together.

3)

Practice: Multiply the radicals.

1)
2)
3)
4)
5)

Squares and Square Roots (A)

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The Squares and Square Roots (A) Math Worksheet Page 1 The Squares and Square Roots (A) Math Worksheet Page 2

FORMATION OF QUADRATIC EQUATION WITH GIVEN ROOTS

The least common multiplication of the denominators 3 and 2 is 6.

Make each denominator as 6 using multiplication.

Formation of quadratic equation : 

x 2  - (sum of the roots)x + product of the roots  =  0 

If one root of a quadratic equation (2 +  √3), then  form the equation given that the roots are irrational.  

(2 +  √3) is an irrational number. 

We already know the fact that irrational roots of a quadratic equation will occur  in conjugate  pairs.

That is,  if  (2 +  √3) is one root of a quadratic equation, then  (2 -  √3) will be  the other root of the same equation. 

So,  (2 +  √3) and  (2 -  √3) are the roots of the required quadratic equation. 

Formation of quadratic equation : 

x 2  - (sum of the roots)x + product of the roots  =  0 

If  α and β be the roots of x 2 + 7x + 12  =  0, find the quadratic equation whose roots are 

Given : α and β be the roots of x 2  + 7x + 12  =  0.

sum of roots  =  -coefficient of x / coefficient of x 2

α + β   =  -7 / 1

product of roots  =  constant term / coefficient of x 2

Quadratic equation with roots  ( α + β) 2 ਊnd (α - β) 2 is

x 2 - [ ( α + β) 2  + (α - β) 2 ]x +  ( α + β) 2 (α - β) 2  =  0 -----(1)

Find the values of  ( α + β) 2 and  ( α - β) 2 .

So, the required quadratic equation is

x 2  - 50 x + 49    =  0

If  α and β be the roots of x 2  + px + q  =  0, find the quadratic equation whose roots are 

Given :  α and β be the roots of x 2  + px + q  =  0.

sum of roots  =  -coefficient of x / coefficient of x 2

product of roots  =  constant term / coefficient of x 2

Quadratic equation with roots  α/βਊnd β/ α   is

So, the required quadratic equation is

Multiply each side by q. 

qx 2  - (p 2  - 2q) x + q    =  0

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ALGEBRA

The four operations and their signs.
The function of parentheses.
Terms versus factors .
Powers and exponents.
The order of operations.
Values and evaluations.
Evaluating algebraic expressions.

The integers.
The algebraic sign and the absolute value.
Subtracting a larger number from a smaller.
The number line.
The negative of any number.

"Adding" a negative number.
Naming terms. The rule for adding terms.
Subtracting a negative number.

The rule of symmetry. Commutative rules. Inverses.
Two rules for equations.

The definition of reciprocals. The definition of division. Rules for 0.

Parentheses. Brackets. Braces.
The relationship of b &minus a to a &minus b .

The law of inverses.
Transposing.
A logical sequence of statements.
Simple fractional equations.

Absolute value equations.
Absolute value inequalities.

Powers of a number.
Rules of exponents: When to add, when to multiply.

The definition of a polynomial in x .
Factoring polynomials.
Factoring by grouping.
Equations in which the unknown is a common factor.

Quadratics in different arguments.

Perfect square trinomials.
The square of a trinomial.
Completing the square.
Geometrical algebra.

Summary of Multiplying/Factoring.
Factoring by grouping.
The sum and difference of any two powers: a n ± b n .

Negative exponents. Exponent 0. Scientific notation.

Rational expressions. The principle of equivalent fractions. Reducing to lowest terms.

The Lowest Common Multiple (LCM) of a series of terms.

The whole is equal to the sum of the parts.
Same time problem: Upstream-downstream.
Total time problem. Job problem.

Square roots.
Equations x ² = a , and the principal square root.
Rationalizing a denominator.
Real numbers.

Roots of numbers. The index of a radical.
Fractional exponents.
Negative exponents.

The square root of a negative number.
The real and imaginary components.
Conjugate pairs.

The distance of a point from the origin.
The distance between any two points.
A proof of the Pythagorean theorem.

The equation of the first degree and its graph.
Vertical and horizontal lines.

The slope intercept form of the equation of a straight line. The general form.
Parallel and perpendicular lines.
The point-slope formula. The two-point formula.

The method of addition. The method of substitution. Cramer's Rule: The method of determinants.
Three equations in three unknowns.

Investment problems. Mixture problems.
Upstream-downstream problems.

The roots of a quadratic.
Solution by factoring.
Completing the square.
The quadratic formula.
The discriminant.
The graph of a quadratic: A parabola.

Definition. The three laws of logarithms.
Common logarithms.

Direct variation. The constant of proportionality.
Varies as the square. Varies inversely. Varies as the inverse square.


Exponents and Roots - GRE

Exponents are used to denote the repeated multiplication of a number by itself.

The following are some rules of exponents. Scroll down the page for more examples and solutions.


For example, 2 4 = 2 × 2 × 2 × 2 = 16
In the expression, 2 4 , 2 is called the base , 4 is called the exponent , and we read the expression as “2 to the fourth power.”

When the exponent is 2, we call the process squaring.

For example,
5 2 = 25, is read as "5 squared is 25".
6 2 = 36, is read as "6 squared is 36".

When negative numbers are raised to powers, the result may be positive or negative. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative.

For example,
(−3) 4 = −3 × −3 × −3 × −3 = 81
(−3) 3 = −3 × −3 × −3 = −27

Take note of the parenthesis: (−3) 2 = 9, but −3 2 = −9

Zero or Negative Exponents

Exponents can also be negative or zero such exponents are defined as follows.

  • For all nonzero numbers a, a 0 = 1.
  • The expression 0 0 is undefined.
  • For all nonzero numbers a,

How to work with zero and negative exponents?

Square Roots

A square root of a nonnegative number n is a number r such that r 2 = n.

For example, 5 is a square root of 25 because 5 2 = 25.

Another square root of 25 is −5 because (−5) 2 is also equals to 25.

The symbol used for square root is . (The symbol is also called the radical sign).

Every positive number a has two square roots: √a, which is positive, and -√a, which is negative.
√16 = 4 and −√16 = −4

The only square root of 0 is 0. Square roots of negative numbers are not defined in the real number system.

Perfect squares and square roots

Some numbers are called perfect squares. It is important we can recognize perfect square when working with square roots.
1 2 = 1
2 2 = 4
3 2 = 9
4 2 = 16
5 2 = 25
6 2 = 36
7 2 = 49
8 2 = 64
9 2 = 81
10 2 = 100

Rules for Square Roots

Here are some important rules regarding operations with square roots, where x > 0 and y > 0


Product Rule & Simplifying Square Roots

Examples:
Simplify
√18
√48
3 √2 ˙ 3 √4
3 √54

Quotient Rule & Simplifying Square Roots

How to use the rules regarding operations with square roots?

Roots of Higher Order

A square root is a root of order 2. Higher-order roots of a positive number n are defined similarly.

The cube root is a root of order 3.

For Example:
8 has one cube root. The cube root of 8 is 2 because 2 3 = 8.
−8 has one cube root. The cube root of −8 is −2 because (−2) 3 = −8

The fourth root is a root of order 4.

For example,
8 has two fourth roots. because 2 4 = 16 and (−2) 4 = 16

These n th roots obey rules similar to the square root.

There are some notable differences between odd order roots and even-order roots (in the real number system):

  • For odd-order roots, there is exactly one root for every number n, even when n is negative. For example, the cube root of 8 is 2 and the cube root of −8 is −2.
  • For even-order roots, there are exactly two roots for every positive number n and no roots for any negative number n. For example, the fourth root of 16 is 2 and −2 and there is no fourth root for −16.

What are radicals and how to simplify perfect square, cube, and nth roots?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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