Properties of Rational Exponents Quick Check

Question

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Question
Solve the following equation with a rational exponent: x23=64
.(1 point)
Responses

32
32

512
512

8
8

16
16
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Answer 1

To solve the equation \( x^{\frac{2}{3}} = 64 \), we can first isolate \( x \).

1. Raise both sides to the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\):

\[ \left( x^{\frac{2}{3}} \right)^{\frac{3}{2}} = 64^{\frac{3}{2}} \]

2. This simplifies to:

\[ x = 64^{\frac{3}{2}} \]

3. Next, calculate \( 64^{\frac{3}{2}} \):
   • First, find \( \sqrt{64} \) (which is \( 8 \)).
   • Then raise it to the power of \( 3 \):

\[ 8^3 = 512 \]

So, \( x = 512 \).

The correct answer is 512.