rational exponents assignment 2

1.3 Radicals and Rational Exponents

Learning objectives.

In this section, you will:

• Evaluate square roots.
• Use the product rule to simplify square roots.
• Use the quotient rule to simplify square roots.
• Add and subtract square roots.
• Rationalize denominators.
• Use rational roots.

A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1 , and use the Pythagorean Theorem.

Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since 4 2 = 16 , 4 2 = 16 , the square root of 16 16 is 4. 4. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.

In general terms, if a a is a positive real number, then the square root of a a is a number that, when multiplied by itself, gives a . a . The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals a . a . The square root obtained using a calculator is the principal square root.

The principal square root of a a is written as a . a . The symbol is called a radical , the term under the symbol is called the radicand , and the entire expression is called a radical expression .

Principal Square Root

The principal square root of a a is the nonnegative number that, when multiplied by itself, equals a . a . It is written as a radical expression , with a symbol called a radical over the term called the radicand : a . a .

Does 25 = ± 5 ? 25 = ± 5 ?

No. Although both 5 2 5 2 and ( −5 ) 2 ( −5 ) 2 are 25 , 25 , the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25 = 5. 25 = 5.

Evaluate each expression.

• ⓒ 25 + 144 25 + 144
• ⓓ 49 − 81 49 − 81
• ⓐ 100 = 10 100 = 10 because 10 2 = 100 10 2 = 100
• ⓑ 16 = 4 = 2 16 = 4 = 2 because 4 2 = 16 4 2 = 16 and 2 2 = 4 2 2 = 4
• ⓒ 25 + 144 = 169 = 13 25 + 144 = 169 = 13 because 13 2 = 169 13 2 = 169
• ⓓ 49 − 81 = 7 − 9 = −2 49 − 81 = 7 − 9 = −2 because 7 2 = 49 7 2 = 49 and 9 2 = 81 9 2 = 81

For 25 + 144 , 25 + 144 , can we find the square roots before adding?

No. 25 + 144 = 5 + 12 = 17. 25 + 144 = 5 + 12 = 17. This is not equivalent to 25 + 144 = 13. 25 + 144 = 13. The order of operations requires us to add the terms in the radicand before finding the square root.

• ⓒ 25 − 9 25 − 9
• ⓓ 36 + 121 36 + 121

Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite 15 15 as 3 ⋅ 5 . 3 ⋅ 5 . We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

The Product Rule for Simplifying Square Roots

If a a and b b are nonnegative, the square root of the product a b a b is equal to the product of the square roots of a a and b . b .

Given a square root radical expression, use the product rule to simplify it.

• Factor any perfect squares from the radicand.
• Write the radical expression as a product of radical expressions.

Simplify the radical expression.

• ⓑ 162 a 5 b 4 162 a 5 b 4
• ⓐ 100 ⋅ 3 Factor perfect square from radicand . 100 ⋅ 3 Write radical expression as product of radical expressions . 10 3 Simplify . 100 ⋅ 3 Factor perfect square from radicand . 100 ⋅ 3 Write radical expression as product of radical expressions . 10 3 Simplify .
• ⓑ 81 a 4 b 4 ⋅ 2 a Factor perfect square from radicand . 81 a 4 b 4 ⋅ 2 a Write radical expression as product of radical expressions . 9 a 2 b 2 2 a Simplify . 81 a 4 b 4 ⋅ 2 a Factor perfect square from radicand . 81 a 4 b 4 ⋅ 2 a Write radical expression as product of radical expressions . 9 a 2 b 2 2 a Simplify .

Simplify 50 x 2 y 3 z . 50 x 2 y 3 z .

Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

• Express the product of multiple radical expressions as a single radical expression.

Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression. 12 ⋅ 3 12 ⋅ 3

12 ⋅ 3 Express the product as a single radical expression . 36 Simplify . 6 12 ⋅ 3 Express the product as a single radical expression . 36 Simplify . 6

Simplify 50 x ⋅ 2 x 50 x ⋅ 2 x assuming x > 0. x > 0.

Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite 5 2 5 2 as 5 2 . 5 2 .

The Quotient Rule for Simplifying Square Roots

The square root of the quotient a b a b is equal to the quotient of the square roots of a a and b , b , where b ≠ 0. b ≠ 0.

Given a radical expression, use the quotient rule to simplify it.

• Write the radical expression as the quotient of two radical expressions.
• Simplify the numerator and denominator.

5 36 Write as quotient of two radical expressions . 5 6 Simplify denominator . 5 36 Write as quotient of two radical expressions . 5 6 Simplify denominator .

Simplify 2 x 2 9 y 4 . 2 x 2 9 y 4 .

Using the Quotient Rule to Simplify an Expression with Two Square Roots

234 x 11 y 26 x 7 y 234 x 11 y 26 x 7 y

234 x 11 y 26 x 7 y Combine numerator and denominator into one radical expression . 9 x 4 Simplify fraction . 3 x 2   Simplify square root . 234 x 11 y 26 x 7 y Combine numerator and denominator into one radical expression . 9 x 4 Simplify fraction . 3 x 2   Simplify square root .

Simplify 9 a 5 b 14 3 a 4 b 5 . 9 a 5 b 14 3 a 4 b 5 .

Adding and Subtracting Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of 2 2 and 3 2 3 2 is 4 2 . 4 2 . However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18 18 can be written with a 2 2 in the radicand, as 3 2 , 3 2 , so 2 + 18 = 2 + 3 2 = 4 2 . 2 + 18 = 2 + 3 2 = 4 2 .

Given a radical expression requiring addition or subtraction of square roots, simplify.

• Simplify each radical expression.
• Add or subtract expressions with equal radicands.

Adding Square Roots

Add 5 12 + 2 3 . 5 12 + 2 3 .

We can rewrite 5 12 5 12 as 5 4 · 3 . 5 4 · 3 . According the product rule, this becomes 5 4 3 . 5 4 3 . The square root of 4 4 is 2, so the expression becomes 5 ( 2 ) 3 , 5 ( 2 ) 3 , which is 10 3 . 10 3 . Now the terms have the same radicand so we can add.

10 3 + 2 3 = 12 3 10 3 + 2 3 = 12 3

Add 5 + 6 20 . 5 + 6 20 .

Subtracting Square Roots

Subtract 20 72 a 3 b 4 c − 14 8 a 3 b 4 c . 20 72 a 3 b 4 c − 14 8 a 3 b 4 c .

Factor 9 out of the first term so that both terms have equal radicands.

Subtract 3 80 x − 4 45 x . 3 80 x − 4 45 x .

Rationalizing Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator .

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.

For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is b c , b c , multiply by c c . c c .

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is a + b c , a + b c , then the conjugate is a − b c . a − b c .

Given an expression with a single square root radical term in the denominator, rationalize the denominator.

• Multiply the numerator and denominator by the radical in the denominator.

Rationalizing a Denominator Containing a Single Term

Write 2 3 3 10 2 3 3 10 in simplest form.

The radical in the denominator is 10 . 10 . So multiply the fraction by 10 10 . 10 10 . Then simplify.

Write 12 3 2 12 3 2 in simplest form.

Given an expression with a radical term and a constant in the denominator, rationalize the denominator.

• Find the conjugate of the denominator.
• Multiply the numerator and denominator by the conjugate.
• Use the distributive property.

Rationalizing a Denominator Containing Two Terms

Write 4 1 + 5 4 1 + 5 in simplest form.

Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of 1 + 5 1 + 5 is 1 − 5 . 1 − 5 . Then multiply the fraction by 1 − 5 1 − 5 . 1 − 5 1 − 5 .

Write 7 2 + 3 7 2 + 3 in simplest form.

Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

Understanding n th Roots

Suppose we know that a 3 = 8. a 3 = 8. We want to find what number raised to the 3rd power is equal to 8. Since 2 3 = 8 , 2 3 = 8 , we say that 2 is the cube root of 8.

The n th root of a a is a number that, when raised to the n th power, gives a . a . For example, −3 −3 is the 5th root of −243 −243 because ( −3 ) 5 = −243. ( −3 ) 5 = −243. If a a is a real number with at least one n th root, then the principal n th root of a a is the number with the same sign as a a that, when raised to the n th power, equals a . a .

The principal n th root of a a is written as a n , a n , where n n is a positive integer greater than or equal to 2. In the radical expression, n n is called the index of the radical.

Principal n n th Root

If a a is a real number with at least one n th root, then the principal n th root of a , a , written as a n , a n , is the number with the same sign as a a that, when raised to the n th power, equals a . a . The index of the radical is n . n .

Simplifying n th Roots

Simplify each of the following:

• ⓐ −32 5 −32 5
• ⓑ 4 4 ⋅ 1 , 024 4 4 4 ⋅ 1 , 024 4
• ⓒ − 8 x 6 125 3 − 8 x 6 125 3
• ⓓ 8 3 4 − 48 4 8 3 4 − 48 4
• ⓐ −32 5 = −2 −32 5 = −2 because ( −2 ) 5 = −32 ( −2 ) 5 = −32
• ⓑ First, express the product as a single radical expression. 4,096 4 = 8 4,096 4 = 8 because 8 4 = 4,096 8 4 = 4,096
• ⓒ − 8 x 6 3 125 3 Write as quotient of two radical expressions . − 2 x 2 5 Simplify . − 8 x 6 3 125 3 Write as quotient of two radical expressions . − 2 x 2 5 Simplify .
• ⓓ 8 3 4 − 2 3 4 Simplify to get equal radicands . 6 3 4   Add . 8 3 4 − 2 3 4 Simplify to get equal radicands . 6 3 4   Add .
• ⓐ −216 3 −216 3
• ⓑ 3 80 4 5 4 3 80 4 5 4
• ⓒ 6 9 , 000 3 + 7 576 3 6 9 , 000 3 + 7 576 3

Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index n n is even, then a a cannot be negative.

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n th root. The numerator tells us the power and the denominator tells us the root.

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

• Rational Exponents

Rational exponents are another way to express principal n th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

Given an expression with a rational exponent, write the expression as a radical.

• Determine the power by looking at the numerator of the exponent.
• Determine the root by looking at the denominator of the exponent.
• Using the base as the radicand, raise the radicand to the power and use the root as the index.

Writing Rational Exponents as Radicals

Write 343 2 3 343 2 3 as a radical. Simplify.

The 2 tells us the power and the 3 tells us the root.

343 2 3 = ( 343 3 ) 2 = 343 2 3 343 2 3 = ( 343 3 ) 2 = 343 2 3

We know that 343 3 = 7 343 3 = 7 because 7 3 = 343. 7 3 = 343. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

343 2 3 = ( 343 3 ) 2 = 7 2 = 49 343 2 3 = ( 343 3 ) 2 = 7 2 = 49

Write 9 5 2 9 5 2 as a radical. Simplify.

Writing Radicals as Rational Exponents

Write 4 a 2 7 4 a 2 7 using a rational exponent.

The power is 2 and the root is 7, so the rational exponent will be 2 7 . 2 7 . We get 4 a 2 7 . 4 a 2 7 . Using properties of exponents, we get 4 a 2 7 = 4 a −2 7 . 4 a 2 7 = 4 a −2 7 .

Write x ( 5 y ) 9 x ( 5 y ) 9 using a rational exponent.

Simplifying Rational Exponents

• ⓐ 5 ( 2 x 3 4 ) ( 3 x 1 5 ) 5 ( 2 x 3 4 ) ( 3 x 1 5 )
• ⓑ ( 16 9 ) − 1 2 ( 16 9 ) − 1 2

ⓐ 30 x 3 4 x 1 5 Multiply the coefficients . 30 x 3 4 + 1 5 Use properties of exponents . 30 x 19 20 Simplify . 30 x 3 4 x 1 5 Multiply the coefficients . 30 x 3 4 + 1 5 Use properties of exponents . 30 x 19 20 Simplify .

ⓑ ( 9 16 ) 1 2    Use definition of negative exponents . 9 16    Rewrite as a radical . 9 16    Use the quotient rule . 3 4    Simplify . ( 9 16 ) 1 2    Use definition of negative exponents . 9 16    Rewrite as a radical . 9 16    Use the quotient rule . 3 4    Simplify .

Simplify ( 8 x ) 1 3 ( 14 x 6 5 ) . ( 8 x ) 1 3 ( 14 x 6 5 ) .

Access these online resources for additional instruction and practice with radicals and rational exponents.

• Simplify Radicals
• Rationalize Denominator

1.3 Section Exercises

What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

Where would radicals come in the order of operations? Explain why.

Every number will have two square roots. What is the principal square root?

Can a radical with a negative radicand have a real square root? Why or why not?

For the following exercises, simplify each expression.

4 ( 9 + 16 ) 4 ( 9 + 16 )

289 − 121 289 − 121

27 64 27 64

169 + 144 169 + 144

18 162 18 162

14 6 − 6 24 14 6 − 6 24

15 5 + 7 45 15 5 + 7 45

96 100 96 100

( 42 ) ( 30 ) ( 42 ) ( 30 )

12 3 − 4 75 12 3 − 4 75

4 225 4 225

405 324 405 324

360 361 360 361

5 1 + 3 5 1 + 3

8 1 − 17 8 1 − 17

128 3 + 3 2 3 128 3 + 3 2 3

−32 243 5 −32 243 5

15 125 4 5 4 15 125 4 5 4

3 −432 3 + 16 3 3 −432 3 + 16 3

400 x 4 400 x 4

4 y 2 4 y 2

( 144 p 2 q 6 ) 1 2 ( 144 p 2 q 6 ) 1 2

m 5 2 289 m 5 2 289

9 3 m 2 + 27 9 3 m 2 + 27

3 a b 2 − b a 3 a b 2 − b a

4 2 n 16 n 4 4 2 n 16 n 4

225 x 3 49 x 225 x 3 49 x

3 44 z + 99 z 3 44 z + 99 z

50 y 8 50 y 8

490 b c 2 490 b c 2

32 14 d 32 14 d

q 3 2 63 p q 3 2 63 p

8 1 − 3 x 8 1 − 3 x

20 121 d 4 20 121 d 4

w 3 2 32 − w 3 2 50 w 3 2 32 − w 3 2 50

108 x 4 + 27 x 4 108 x 4 + 27 x 4

12 x 2 + 2 3 12 x 2 + 2 3

147 k 3 147 k 3

125 n 10 125 n 10

42 q 36 q 3 42 q 36 q 3

81 m 361 m 2 81 m 361 m 2

72 c − 2 2 c 72 c − 2 2 c

144 324 d 2 144 324 d 2

24 x 6 3 + 81 x 6 3 24 x 6 3 + 81 x 6 3

162 x 6 16 x 4 4 162 x 6 16 x 4 4

64 y 3 64 y 3

128 z 3 3 − −16 z 3 3 128 z 3 3 − −16 z 3 3

1,024 c 10 5 1,024 c 10 5

Real-World Applications

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating 90,000 + 160,000 . 90,000 + 160,000 . What is the length of the guy wire?

A car accelerates at a rate of 6 − 4 t m/s 2 6 − 4 t m/s 2 where t is the time in seconds after the car moves from rest. Simplify the expression.

8 − 16 4 − 2 − 2 1 2 8 − 16 4 − 2 − 2 1 2

4 3 2 − 16 3 2 8 1 3 4 3 2 − 16 3 2 8 1 3

m n 3 a 2 c −3 ⋅ a −7 n −2 m 2 c 4 m n 3 a 2 c −3 ⋅ a −7 n −2 m 2 c 4

a a − c a a − c

x 64 y + 4 y 128 y x 64 y + 4 y 128 y

( 250 x 2 100 b 3 ) ( 7 b 125 x ) ( 250 x 2 100 b 3 ) ( 7 b 125 x )

64 3 + 256 4 64 + 256 64 3 + 256 4 64 + 256

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• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/1-3-radicals-and-rational-exponents

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1.2.19: Simplifying Rational Exponents

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• Page ID 128496

• Katherine Skelton
• Highline College

Simplifying Expressions with \(a^{\frac{1}{n}}\)

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that \(\left(a^{m}\right)^{n}=a^{m \cdot n}\) when \(m\) and \(n\) are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number \(p\) such that \(\left(8^{p}\right)^{3}=8\). We will use the Power Property of Exponents to find the value of \(p\).

\(\left(8^{p}\right)^{3}=8\)

Multiple the exponents on the left.

\(8^{3p}=8\)

Write the exponent \(1\) on the right.

\(8^{3p}=8^{1}\)

Since the bases are the same, the exponents must be equal.

Solve for \(p\).

\(p=\frac{1}{3}\)

So \(\left(8^{\frac{1}{3}}\right)^{3}=8\). But we know also \((\sqrt[3]{8})^{3}=8\). Then it must be that \(8^{\frac{1}{3}}=\sqrt[3]{8}\).

This same logic can be used for any positive integer exponent \(n\) to show that \(a^{\frac{1}{n}}=\sqrt[n]{a}\).

Rational Exponent \(a^{\frac{1}{n}}\)

If \(\sqrt[n]{a}\) is a real number and \(n \geq 2\), then

\(a^{\frac{1}{n}}=\sqrt[n]{a}\)

The denominator of the rational exponent is the index of the radical.

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you'll practice converting expressions between these two notations.

Example \(\PageIndex{1}\)

Write as a radical expression:

\(x^{\frac{1}{2}}\)

\(y^{\frac{1}{3}}\)

\(z^{\frac{1}{4}}\)

We want to write each expression in the form \(\sqrt[n]{a}\).

The denominator of the rational exponent is \(2\), so the index of the radical is \(2\). We do not show the index when it is \(2\).

\(\sqrt{x}\)

The denominator of the exponent is \(3\), so the index is \(3\).

\(\sqrt[3]{y}\)

The denominator of the exponent is \\(4\), so the index is \(4\).

\(\sqrt[4]{z}\)

You Try \(\PageIndex{1}\)

• \(t^{\frac{1}{2}}\)
• \(m^{\frac{1}{3}}\)
• \(r^{\frac{1}{4}}\)
• \(\sqrt{t}\)
• \(\sqrt[3]{m}\)
• \(\sqrt[4]{r}\)

In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.

Example \(\PageIndex{2}\)

Write with a rational exponent:

\(\sqrt{5y}\)

\(\sqrt[3]{4 x}\)

\(3 \sqrt[4]{5 z}\)

We want to write each radical in the form \(a^{\frac{1}{n}}\)

No index is shown, so it is \(2\).

The denominator of the exponent will be \(2\).

Put parentheses around the entire expression \(5y\).

\((5 y)^{\frac{1}{2}}\)

The index is \(3\), so the denominator of the exponent is \(3\). Include parentheses \((4x)\).

\((4 x)^{\frac{1}{3}}\)

The index is \(4\), so the denominator of the exponent is \(4\). Put parentheses only around the \(5z\) since 3 is not under the radical sign.

\(3(5 z)^{\frac{1}{4}}\)

You Try \(\PageIndex{2}\)

• \(\sqrt[7]{3 k}\)
• \(\sqrt[4]{5 j}\)
• \(8 \sqrt[3]{2 a}\)
• \((3 k)^{\frac{1}{7}}\)
• \((5 j)^{\frac{1}{4}}\)
• \(8(2 a)^{\frac{1}{3}}\)

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Example \(\PageIndex{3}\)

\(25^{\frac{1}{2}}\)

\(64^{\frac{1}{3}}\)

\(256^{\frac{1}{4}}\)

Rewrite as a square root.

\(\sqrt{25}\)

Rewrite as a cube root.

\(\sqrt[3]{64}\)

Recognize \(64\) is a perfect cube.

\(\sqrt[3]{4^{3}}\)

Rewrite as a fourth root.

\(\sqrt[4]{256}\)

Recognize \(256\) is a perfect fourth power.

\(\sqrt[4]{4^{4}}\)

You Try \(\PageIndex{3}\)

• \(36^{\frac{1}{2}}\)
• \(8^{\frac{1}{3}}\)
• \(16^{\frac{1}{4}}\)

Be careful of the placement of the negative signs in the next example. We will need to use the property \(a^{-n}=\frac{1}{a^{n}}\) in one case.

Example \(\PageIndex{4}\)

\((-16)^{\frac{1}{4}}\)

\(-16^{\frac{1}{4}}\)

\((16)^{-\frac{1}{4}}\)

\(\sqrt[4]{-16}\)

There is no real number raised to the fourth power that gives us -16 so this is not a real number.

The exponent only applies to the \(16\). Rewrite as a fourth root.

\(-\sqrt[4]{16}\)

Rewrite \(16\) as \(2^{4}\)

\(-\sqrt[4]{2^{4}}\)

Rewrite using the property \(a^{-n}=\frac{1}{a^{n}}\).

\(\frac{1}{(16)^{\frac{1}{4}}}\)

\(\frac{1}{\sqrt[4]{16}}\)

Rewrite \(16\) as \(2^{4}\).

\(\frac{1}{\sqrt[4]{2^{4}}}\)

\(\frac{1}{2}\)

You Try \(\PageIndex{4}\)

• \((-64)^{-\frac{1}{2}}\)
• \(-64^{\frac{1}{2}}\)
• \((64)^{-\frac{1}{2}}\)
• Not a real number
• \(\frac{1}{8}\)

Simplifying Expressions with \(a^{\frac{m}{n}}\)

We can look at \(a^{\frac{m}{n}}\) in two ways. Remember the Power Property tells us to multiply the exponents and so \(\left(a^{\frac{1}{n}}\right)^{m}\) and \(\left(a^{m}\right)^{\frac{1}{n}}\) both equal \(a^{\frac{m}{n}}\). If we write these expressions in radical form, we get

\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}}\)

This leads us to the following definition.

Rational Exponent \(a^{\frac{m}{n}}\)

For any positive integers \(m\) and \(n\),

\(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\)

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.

Example \(\PageIndex{5}\)

• \(\sqrt{y^{3}}\)
• \((\sqrt[3]{2 x})^{4}\)
• \(\sqrt{\left(\frac{3 a}{4 b}\right)^{3}}\)

We want to use \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) to write each radical in the form \(a^{\frac{m}{n}}\)

Figure \(\PageIndex{1}\)

Figure \(\PageIndex{2}\)

Figure \(\PageIndex{3}\)

You Try \(\PageIndex{5}\)

• \(\sqrt[5]{a^{2}}\)
• \((\sqrt[3]{5 a b})^{5}\)
• \(\sqrt{\left(\frac{7 x y}{z}\right)^{3}}\)
• \(a^{\frac{2}{5}}\)
• \((5 a b)^{\frac{5}{3}}\)
• \(\left(\frac{7 x y}{z}\right)^{\frac{3}{2}}\)

Remember that \(a^{-n}=\frac{1}{a^{n}}\). The negative sign in the exponent does not change the sign of the expression.

Example \(\PageIndex{6}\)

\(125^{\frac{2}{3}}\)

\(16^{-\frac{3}{2}}\)

\(32^{-\frac{2}{5}}\)

We will rewrite the expression as a radical first using the defintion, \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}\). This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

The power of the radical is the numerator of the exponent, \(2\). The index of the radical is the denominator of the exponent, \(3\).

\((\sqrt[3]{125})^{2}\)

\((5)^{2}\)

b. We will rewrite each expression first using \(a^{-n}=\frac{1}{a^{n}}\) and then change to radical form.

Rewrite using \(a^{-n}=\frac{1}{a^{n}}\)

\(\frac{1}{16^{\frac{3}{2}}}\)

Change to radical form. The power of the radical is the numerator of the exponent, \(3\). The index is the denominator of the exponent, \(2\).

\(\frac{1}{(\sqrt{16})^{3}}\)

\(\frac{1}{4^{3}}\)

\(\frac{1}{64}\)

\(\frac{1}{32^{\frac{2}{5}}}\)

Change to radical form.

\(\frac{1}{(\sqrt[5]{32})^{2}}\)

Rewrite the radicand as a power.

\(\frac{1}{\left(\sqrt[5]{2^{5}}\right)^{2}}\)

\(\frac{1}{2^{2}}\)

\(\frac{1}{4}\)

You Try \(\PageIndex{6}\)

• \(4^{\frac{3}{2}}\)
• \(27^{-\frac{2}{3}}\)
• \(625^{-\frac{3}{4}}\)
• \(\frac{1}{9}\)
• \(\frac{1}{125}\)

Example \(\PageIndex{7}\)

\(-25^{\frac{3}{2}}\)

\(-25^{-\frac{3}{2}}\)

\((-25)^{\frac{3}{2}}\)

Rewrite in radical form.

\(-(\sqrt{25})^{3}\)

Simplify the radical.

\(-(5)^{3}\)

Rewrite using \(a^{-n}=\frac{1}{a^{n}}\).

\(-\left(\frac{1}{25^{\frac{3}{2}}}\right)\)

\(-\left(\frac{1}{(\sqrt{25})^{3}}\right)\)

\(-\left(\frac{1}{(5)^{3}}\right)\)

\(-\frac{1}{125}\)

\((\sqrt{-25})^{3}\)

There is no real number whose square root is \(-25\).

Not a real number.

You Try \(\PageIndex{7}\)

• \(-16^{\frac{3}{2}}\)
• \(-16^{-\frac{3}{2}}\)
• \((-16)^{-\frac{3}{2}}\)
• \(-\frac{1}{64}\)
• Not a real number

Using the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponents here to have them for reference as we simplify expressions.

Properties of Exponents

If \(a\) and \(b\) are real numbers and \(m\) and \(n\) are rational numbers, then

Product Property

\(a^{m} \cdot a^{n}=a^{m+n}\)

Power Property

\(\left(a^{m}\right)^{n}=a^{m \cdot n}\)

Product to a Power

\((a b)^{m}=a^{m} b^{m}\)

Quotient Property

\(\frac{a^{m}}{a^{n}}=a^{m-n}, a \neq 0\)

Zero Exponent Definition

\(a^{0}=1, a \neq 0\)

Quotient to a Power Property

\(\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}}, b \neq 0\)

Negative Exponent Property

\(a^{-n}=\frac{1}{a^{n}}, a \neq 0\)

We will apply these properties in the next example.

Example \(\PageIndex{8}\)

\(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\)

\(\left(z^{9}\right)^{\frac{2}{3}}\)

\(\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\)

a. The Product Property tells us that when we multiple the same base, we add the exponents.

The bases are the same, so we add the exponents.

\(x^{\frac{1}{2}+\frac{5}{6}}\)

Add the fractions.

\(x^{\frac{8}{6}}\)

Simplify the exponent.

\(x^{\frac{4}{3}}\)

b. The Power Property tells us that when we raise a power to a power, we multiple the exponents.

To raise a power to a power, we multiple the exponents.

\(z^{9 \cdot \frac{2}{3}}\)

c. The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

To divide with the same base, we subtract the exponents.

\(\frac{1}{x^{\frac{5}{3}-\frac{1}{3}}}\)

\(\frac{1}{x^{\frac{4}{3}}}\)

You Try \(\PageIndex{8}\)

• \(y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}\)
• \(\left(m^{9}\right)^{\frac{2}{9}}\)
• \(\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}\)
• \(y^{\frac{11}{8}}\)
• \(\frac{1}{d}\)

Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property .

Example \(\PageIndex{9}\)

\(\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

\(\left(m^{\frac{2}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

First we use the Product to a Power Property.

\((27)^{\frac{2}{3}}\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

Rewrite \(27\) as a power of \(3\).

\(\left(3^{3}\right)^{\frac{2}{3}}\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

\(\left(3^{2}\right)\left(u^{\frac{1}{3}}\right)\)

\(9 u^{\frac{1}{3}}\)

\(\left(m^{\frac{2}{3}}\right)^{\frac{3}{2}}\left(n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

To raise a power to a power, we multiply the exponents.

\(m n^{\frac{3}{4}}\)

You Try \(\PageIndex{9}\)

• \(\left(81 n^{\frac{2}{5}}\right)^{\frac{3}{2}}\)
• \(\left(a^{\frac{3}{2}} b^{\frac{1}{2}}\right)^{\frac{4}{3}}\)
• \(729 n^{\frac{3}{5}}\)
• \(a^{2} b^{\frac{2}{3}}\)

We will use both the Product Property and the Quotient Property in the next example.

Example \(\PageIndex{10}\)

\(\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}\)

\(\left(\frac{16 x^{\frac{4}{3}} y^{-\frac{5}{6}}}{x^{-\frac{2}{3}} y^{\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Use the Product Property in the numerator, add the exponents.

\(\frac{x^{\frac{2}{4}}}{x^{-\frac{6}{4}}}\)

Use the Quotient Property, subtract the exponents.

\(x^{\frac{8}{4}}\)

\(\left(\frac{16 x^{\frac{6}{3}}}{y^{\frac{6}{6}}}\right)^{\frac{1}{2}}\)

\(\left(\frac{16 x^{2}}{y}\right)^{\frac{1}{2}}\)

Use the Product to a Power Property, multiply the exponents.

\(\frac{4 x}{y^{\frac{1}{2}}}\)

You Try \(\PageIndex{10}\)

• \(\frac{m^{\frac{2}{3}} \cdot m^{-\frac{1}{3}}}{m^{-\frac{5}{3}}}\)
• \(\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{-\frac{1}{6}}}\right)^{\frac{1}{2}}\)
• \(\frac{5 n}{m^{\frac{1}{4}}}\)

Sam Richardson showcases Detroit hot spots and swagger in ad campaign timed to NFL draft

Sam Richardson has probably done more for Detroit than most full-time goodwill ambassadors. Now the star of “Detroiters” is again helping show off his hometown with a promotional campaign timed to the NFL draft, happening here Thursday through Saturday.

In a roughly six-minute video from Pure Michigan, the Michigan Economic Development Corporation and Visit Detroit, he takes viewers on a journey to five pretty amazing locations: Hamtramck's Planet Ant Theatre (where he got his start as a performer) and Fowling Warehouse (home to a football/bowling pin game), the record shop/wine bar combo Paramita Sound, the iconic Lafayette Coney Island and Hot Sam’s, the oldest men’s clothing store in downtown Detroit.

Richardson shares the spotlight with a few special guests, including Detroit Lions great Barry Sanders. In addition to the main video, shorter spots will be running on Amazon, ABC, Barstool Sports, ESPN, Meta and TikTok through May 22.

It sounds like a perfect assignment for Richardson, who won an Emmy this year for his guest turn on AppleTV+’s “Ted Lasso.."

”Anybody who’s from Detroit talks about Detroit, we love it so much,” says Richardson, who’s now based in Los Angeles. “It’s nice to do it in a semi-official capacity, but I’d do it anyway.”

Richardson's credits include co-starring in HBO’s “Veep” and AppleTV+’s “The Afterparty" and movies like "Hocus Pocus 2." But his key credit locally is the much-missed Comedy Central series "Detroiters." He and his real-life best friend, Tim Robinson (who grew up in the Motor City's suburbs), created and starred in the critically acclaimed show for two seasons that were filmed here and saturated with Detroit references.

During a conversation via Zoom that has been edited for space and clarity, Richardson spoke to the Free Press about the new campaign, his status as a Lions superfan, his feelings about onions on coney dogs and more.

Question: The video explores Detroit swagger. For those who may not be Detroiters, what is Detroit swagger and how do you get it?

Answer: Detroit swagger, it is hard to exactly describe. You know it when you feel it and you see it. Detroit has its own unique sense of style, and that goes for modern style and historically. We shot at Hot Sam’s, the oldest clothier in Detroit. Looking at the style of clothing, Detroit is known for very specific cuts of suits. Also we’re known for monochromatic suits, gators, all that. I guess I’ve described it by not describing it very well.  How do you get it? You’re born with it. How do you have it? You have it by being there.

Q: There’s a great moment at Hot Sam’s where you ask Barry Sanders about style musts for Detroit football players and he replies, “Something like this,” referring to what he’s trying on. He’s got it, right?

A:  He was swagged out! It was so wild to get to do that with him, too. I was out of my head. His look for that shoot, my look for that shoot, we were both looking real good. It was an all-time moment for me.

Q: GQ called you and Tim Robinson “celebrity psycho fans” for the Detroit Lions playoff run. How psycho? How much were you guys into it?

A: I think a psycho never realizes they’re a psycho. Their thinking is always this is normal, this is completely rational. I think everything I did was completely rational. It was the most exciting season of sports, period, of my adult life. It’s been amazing. To get the invite to come to the games and stuff, there was no other place I’d rather be in the world.

I was in Detroit for the 2023 season opener when the Lions were playing Kansas City, and I was at Tin Roof right by Comerica and Ford Field, and everyone was full of electricity watching us beat the Kansas City Chiefs , last season’s champions. You could see, everybody was like, ‘Uh-oh, this is it! This is the year!’ To watch this team do so well, it’s so rewarding. It ignites the psycho fandom  in me. It’s always there, though. But it’s nice having the world jump on board, too.

Q: To see the rest of the country playing attention to the Detroit Lions, that was mind-blowing.

A: Everywhere, people were like, "Go, Lions!" Anybody I talked to, they were like, "Oh, we were really hoping for the Lions!" It’s not that this is the first year we’ve been good. But this is the first year we’ve had the full respect of everybody. I’m so happy that Detroit has taken the moniker of America’s team. The Lions and the city of Detroit, all this is truly what the American dream or ideal is. The team is an embodiment of the city. The city is an embodiment of America.

Q: Regarding your stop at Lafayette Coney Island, are the onions something you really like on your coney dog or was that for the camera?

A: I love the onions. That’s legit. Its an important part of the dog experience.

Q: Was Lafayette Coney Island an important place for you in your early comedy days after a show? Would that fit into your routine?

A:  Hamtramck (where Planet Ant is located) is a little bit of a hike. But whenever I’m downtown, I do go.  My dad used to have an office in a building right across from it, so I would go when I was 6 and 7 all the time. The Lafayette is truly engrained in me as part of my life and part of my tastebuds. And now I’ve got a signed picture there. I just won an Emmy last year and, maybe, it's better than that.

Q: Thank you for showing the spinning and the sipping at Parmita Sound. Do you remember the first vinyl record you ever bought?

A: I had a vinyl collection growing up. They weren’t new. They were always used or found. I had the “Superman” soundtrack, and I had, like, the “Superman” story on vinyl. I had a "Sesame Street" vinyl. "Sesame Street" disco is actually what it was. It featured songs from Cookie Monster in disco, a great hit, still a legit banger. And the "Rocky IV" soundtrack. And we had a lot of Motown records. Those weren’t my own,  but I’d play them.

More: Your guide to 2024 NFL draft in Detroit: Registration, map, parking, things to do and more

Q: Congratulations on your Emmy for playing Edwin Akufo on “Ted Lasso," a billionaire who’s attempting to buy a soccer team. If through a twist of fate, you became a giant tech billionaire or something, would you pursue buying a professional sports team in Detroit?

A: I would, I would, I would. Just for the good tickets really. I would, hands down.

Q: The Emmys that you received for “Ted Lasso” and that Tim Robinson received for Netflix's “I Think You Should Leave,” I think we all sort of took them as a win for Detroit.

A: That’s what it felt like for me. I will always attribute any success that I have in my life to Detroit., because it’s where I got my start. It’s where my personality got baked and cooked, and my comedy, my comedic voice. It’s where I learned to improvise. That happened here. I’m saying here as if I’m in Detroit, because I’m always talking about Detroit as “here.” I don’t live in Detroit now, because I need to live in L.A. for work. But any chance I get to come home, I take it.