Question
Properties of Rational Exponents Quick Check
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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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Answers
GPT-4o mini
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**.
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**.