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All courses Math II · N-RN.1 60 of 73
Explain rational exponents as extensions of integer exponent rules and as radical notation.

Rewrite an expression with exponent \(1/n\) in radical notation

Problem
Use a¹/ⁿ=ⁿ√a with n=2 to rewrite a¹/² in radical notation.
Your answer
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Hint

Use the denominator of the exponent to name the root.

An exponent of \(1/n\) means the \(n\)th root of the base.

Solution walkthrough

01

Use the rational-exponent definition

\[a^(1/n)=\text{nth}-\text{root}(a)\]

The denominator n of a unit fraction names the root index.

02

Substitute the given index

\[n=2~->~a^(1/2)=2\text{nd}-\text{root}(a)\]

The prompt explicitly supplies n=2.

03

Use square-root notation

\[2\text{nd}-\text{root}(a)=\text{sqrt}(a)\]

By convention the index 2 is omitted from a radical sign.

04

Check by squaring

\[(\text{sqrt}(a))^2=a\]

On the domain where the principal square root is defined, squaring reverses the square-root operation.

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Another way

  1. Ask for the value whose second power is a; that is exactly the square root of a.

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Common mistake

Do not interpret the denominator 2 as a power of 2. It specifies a square root.