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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 6

Decompose Fractional Powers

Use exponent laws to rewrite exponents 1/2 and 1/3 as roots and generate equivalent expressions.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Split the fraction in the exponent

\(a^{\frac{1}{2}}\) is a square root. \(a^{\frac{2}{2}}\) = a. A numerator of 1 with denominator 2 or 3 is a root.

Power of a power

(\(a^{\frac{1}{2}}\))² = a. (\(a^{\frac{1}{3}}\))³ = a.

Rewrite a root as a power

That lets you use exponent laws on radicals.

Stay with 2 and 3 in the denominator

This page does not use other denominators.

Level 1

Easy

Rewrite one way, then the other.

Example
Rewrite \(5^{\frac{1}{2}}\) as a radical, then rewrite √5 as a rational exponent.

√5

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

Answer: √5 = 5^(1/2)

1
Rewrite \(8^{\frac{1}{3}}\) as a radical and simplify.
Show Answer

∛8 = 2

Answer: 2

2
Rewrite √9 as a rational exponent and simplify.
Show Answer

\(9^{\frac{1}{2}}\) = 3

Answer: 3

3
What is (\(7^{\frac{1}{2}}\))²?
Show Answer

7

Answer: 7

4
What is (\(4^{\frac{1}{3}}\))³?
Show Answer

4

Answer: 4

5
Rewrite ∛27 using exponent 13.
Show Answer

\(27^{\frac{1}{3}}\) = 3

Answer: 3

Level 2

Medium

Use a law of exponents after you rewrite.

Example
Simplify \(16^{\frac{1}{2}}\) × \(16^{\frac{1}{2}}\) using exponents, then using radicals.

\(16^{\frac{1+1}{2}}\) = \(16^{\frac{2}{2}}\) = 16

√16 × √16 = 4 × 4 = 16

Answer: 16

1
Simplify \(8^{\frac{1}{3}}\)\(8^{\frac{1}{3}}\).
Show Answer

The exponents subtract to 0.

8⁰ = 1

Answer: 1

2
Rewrite √a × √a with exponents, for a ≥ 0.
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\(a^{\frac{1}{2}}\) × \(a^{\frac{1}{2}}\) = a

Answer: a

3
Show ∛b × ∛b × ∛b = b using 13.
Show Answer

Three factors of \(b^{\frac{1}{3}}\) add to exponent 1.

Answer: b

4
Simplify (√5)² two ways.
Show Answer

5

Or \(5^{\frac{2}{2}}\) = 5

Answer: 5

5
Rewrite 1√x as a rational exponent, x > 0.
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\(x^{\frac{-1}{2}}\) or 1\(x^{\frac{1}{2}}\)

Answer: x^(−1/2)

Level 3

Hard

Decompose a fractional power and generate an equivalent radical.

Example
Decompose \(8^{\frac{1}{3}}\) as a cube root, then show (\(8^{\frac{1}{3}}\))³ = 8 two ways.

\(8^{\frac{1}{3}}\) = ∛8 = 2

2³ = 8

Exponent: \(8^{\frac{3}{3}}\) = 8

Answer: 8

1
Write three equivalent expressions for the cube root of 8.
Show Answer

∛8

\(8^{\frac{1}{3}}\)

2

Answer: ∛8 = 8^(1/3) = 2

2
Simplify \(36^{\frac{1}{2}}\) × \(4^{\frac{1}{2}}\).
Show Answer

6 × 2 = 12

Or √36 × √4

Answer: 12

3
Why is \((-8)^{\frac{1}{2}}\) not real, while \((-8)^{\frac{1}{3}}\) is?
Show Answer

12 is a square root. Square roots of negatives are not real here.

13 is a cube root. ∛−8 = −2.

Answer: cube root is real

4
Rewrite √x³ for x ≥ 0 using 12, then as x√x.
Show Answer

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

x × \(x^{\frac{1}{2}}\)

Answer: x√x

5
Show \(27^{\frac{1}{3}}\)\(8^{\frac{1}{3}}\) = 32.
Show Answer

∛27∛8 = 32

Answer: 32

Super Challenge

Super Challenge

1
Starting from \(a^{\frac{1}{2}}\), generate three equivalent expressions: a radical, the product of two copies, and the square of the rational exponent.
Show Answer

√a

√a × √a = a

(\(a^{\frac{1}{2}}\))² = a

Answer: √a; product a; square a

2
Use exponent laws to explain why ∛a³ = a when a is a real number.
Show Answer

\(a^{\frac{3}{3}}\) = a¹ = a

The index 3 and the exponent 3 cancel.

Answer: a

3
Write √8 as \(8^{\frac{1}{2}}\) and as 2\(2^{\frac{1}{2}}\). Justify the second form with exponent laws.
Show Answer

8 = 4 × 2

\(8^{\frac{1}{2}}\) = \(4^{\frac{1}{2}}\) × \(2^{\frac{1}{2}}\) = 2√2

Answer: 2√2

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