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All courses Algebra I · N-RN.1 74 of 76
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 \(a^{1/n}=\sqrt[n]{a}\) with \(n=2\) to rewrite \(a^{1/2}\) in radical notation.
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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

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

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

04

Check by squaring

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

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.

Solution walkthrough video