Question
Solve the following equation with a rational exponent: x2/3=64 .(1 point)
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Answered by
GPT-4o mini
To solve the equation \( x^{2/3} = 64 \), follow these steps:
1. **Raise both sides to the reciprocal of the exponent**: The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
\[
(x^{2/3})^{\frac{3}{2}} = 64^{\frac{3}{2}}
\]
2. **Simplify the left side**:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]
3. **Calculate the right side**:
First, calculate \( 64^{1/2} \):
\[
64^{1/2} = \sqrt{64} = 8
\]
Now raise that result to the power of 3:
\[
64^{3/2} = 8^3 = 512
\]
4. **Write the final equation**:
\[
x = 512
\]
Thus, the solution to the equation \( x^{2/3} = 64 \) is
\[
\boxed{512}
\]
1. **Raise both sides to the reciprocal of the exponent**: The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
\[
(x^{2/3})^{\frac{3}{2}} = 64^{\frac{3}{2}}
\]
2. **Simplify the left side**:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]
3. **Calculate the right side**:
First, calculate \( 64^{1/2} \):
\[
64^{1/2} = \sqrt{64} = 8
\]
Now raise that result to the power of 3:
\[
64^{3/2} = 8^3 = 512
\]
4. **Write the final equation**:
\[
x = 512
\]
Thus, the solution to the equation \( x^{2/3} = 64 \) is
\[
\boxed{512}
\]
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