Asked by lolbit
Solve the following equation with a rational exponent: x 2/3 = 64
512
32
16
8
512
32
16
8
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To solve the equation \( x^{2/3} = 64 \), we can follow these steps:
1. **Isolate \( x \)**: To eliminate the exponent \( \frac{2}{3} \), we can raise both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
\[
(x^{2/3})^{3/2} = 64^{3/2}
\]
2. **Simplify the left side**: Using the property of exponents, we have:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]
3. **Calculate the right side**: Now we need to calculate \( 64^{3/2} \).
- First, we can find the square root of \( 64 \):
\[
\sqrt{64} = 8
\]
- Next, we raise \( 8 \) to the power of \( 3 \):
\[
8^3 = 512
\]
4. **Combine the results**: Now we have:
\[
x = 512
\]
Thus, the solution to the equation \( x^{2/3} = 64 \) is \( x = 512 \).
The correct answer from the options provided is \( 512 \).
1. **Isolate \( x \)**: To eliminate the exponent \( \frac{2}{3} \), we can raise both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
\[
(x^{2/3})^{3/2} = 64^{3/2}
\]
2. **Simplify the left side**: Using the property of exponents, we have:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]
3. **Calculate the right side**: Now we need to calculate \( 64^{3/2} \).
- First, we can find the square root of \( 64 \):
\[
\sqrt{64} = 8
\]
- Next, we raise \( 8 \) to the power of \( 3 \):
\[
8^3 = 512
\]
4. **Combine the results**: Now we have:
\[
x = 512
\]
Thus, the solution to the equation \( x^{2/3} = 64 \) is \( x = 512 \).
The correct answer from the options provided is \( 512 \).
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