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All courses Math II · N-RN.2 61 of 73
Rewrite expressions with radicals and rational exponents using exponent properties.

Simplify a product of like bases by adding rational exponents

Problem
For x≥0, simplify x¹/²·x³/² by adding the exponents on the common base. Preserve the stated domain in the result.
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Hint

Because the base is the same, add the exponents \(1/2\) and \(3/2\).

For matching bases, \(b^m \cdot b^n = b^{m+n}\), even when the exponents are fractions.

Solution walkthrough

01

Confirm a common base

\[x^(1/2)*x^(3/2)\]

Both factors use base x, so the product rule for exponents applies.

02

Add the exponents

\[x^(1/2+3/2)\]

Multiplying like bases keeps the base and adds their exponents.

03

Compute the exponent sum

\[1/2+3/2=4/2=2~->~x^2\]

The denominators already match, so add numerators and reduce.

04

Preserve and check the domain

\[x^2,~\text{for}~x\ge~0\]

The original statement restricts x to nonnegative values, and the simplified result must report that stated domain.

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Another way

  1. For x>=0, sqrt(x)*x*sqrt(x)=x*(sqrt(x))²=x².

!

Common mistake

Do not multiply the exponents. Like-base multiplication uses addition of exponents.