Question
solve the equation with a rational exponent: x2/3 = 64
32
512
16
8
32
512
16
8
Answers
GPT-4o mini
To solve the equation \( x^{2/3} = 64 \), we first need to isolate \( x \).
1. 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. This simplifies to:
\[
x^{(2/3) \cdot (3/2)} = 64^{3/2}
\]
\[
x^1 = 64^{3/2}
\]
3. Now we calculate \( 64^{3/2} \). First, find \( 64^{1/2} \):
\[
64^{1/2} = 8
\]
4. Then, compute \( 8^3 \):
\[
8^3 = 512
\]
5. Thus, we have:
\[
x = 512
\]
So the solution to the equation \( x^{2/3} = 64 \) is \( \boxed{512} \).
1. 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. This simplifies to:
\[
x^{(2/3) \cdot (3/2)} = 64^{3/2}
\]
\[
x^1 = 64^{3/2}
\]
3. Now we calculate \( 64^{3/2} \). First, find \( 64^{1/2} \):
\[
64^{1/2} = 8
\]
4. Then, compute \( 8^3 \):
\[
8^3 = 512
\]
5. Thus, we have:
\[
x = 512
\]
So the solution to the equation \( x^{2/3} = 64 \) is \( \boxed{512} \).