To solve the equation [tex]\(x^{\frac{2}{3}} = 64\)[/tex], we need to isolate [tex]\(x\)[/tex]. Start by raising both sides of the equation to the power that will eliminate the exponent on [tex]\(x\)[/tex]. In this case, the exponent is [tex]\(\frac{2}{3}\)[/tex], so we raise both sides to the power of [tex]\(\frac{3}{2}\)[/tex].
Here are the steps:
1. Write down the original equation:
[tex]\[
x^{\frac{2}{3}} = 64
\][/tex]
2. Raise both sides of the equation to the power of [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
\left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}}
\][/tex]
3. Simplify the left side of the equation. When you raise a power to a power, you multiply the exponents:
[tex]\[
x^{\left(\frac{2}{3} \cdot \frac{3}{2}\right)} = x^1 = x
\][/tex]
4. Now, calculate the right side of the equation:
[tex]\[
64^{\frac{3}{2}}
\][/tex]
5. The operation [tex]\(64^{\frac{3}{2}}\)[/tex] can be split into two steps:
- First, find the square root of 64:
[tex]\[
\sqrt{64} = 8
\][/tex]
- Then, cube the result:
[tex]\[
8^3 = 512
\][/tex]
So [tex]\(64^{\frac{3}{2}} = 512\)[/tex].
Therefore:
[tex]\[
x = 512
\][/tex]
The solution to the equation [tex]\(x^{\frac{2}{3}} = 64\)[/tex] is [tex]\(x = 512\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{512}
\][/tex]
