To solve the equation [tex]\( x^{\frac{2}{3}} = 64 \)[/tex], let's follow the steps meticulously:
1. Understand the equation: We are given [tex]\( x^{\frac{2}{3}} = 64 \)[/tex]. The goal is to solve for [tex]\( x \)[/tex].
2. Eliminate the rational exponent: To isolate [tex]\( x \)[/tex], we can raise both sides of the equation to the reciprocal of [tex]\( \frac{2}{3} \)[/tex], which is [tex]\( \frac{3}{2} \)[/tex]. This way, we cancel out the exponent on the left-hand side.
[tex]\[
\left( x^{\frac{2}{3}} \right)^{\frac{3}{2}} = 64^{\frac{3}{2}}
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
x^{\left(\frac{2}{3} \cdot \frac{3}{2}\right)} = 64^{\frac{3}{2}}
\][/tex]
Since [tex]\( \frac{2}{3} \cdot \frac{3}{2} = 1 \)[/tex], we have:
[tex]\[
x = 64^{\frac{3}{2}}
\][/tex]
3. Evaluate the right-hand side: Now, compute [tex]\( 64^{\frac{3}{2}} \)[/tex]:
[tex]\[
64^{\frac{3}{2}} = \left( 64^{\frac{1}{2}} \right)^3
\][/tex]
First, find [tex]\( 64^{\frac{1}{2}} \)[/tex]:
[tex]\[
64^{\frac{1}{2}} = \sqrt{64} = 8
\][/tex]
Now raise 8 to the power of 3:
[tex]\[
8^3 = 512
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 512 \)[/tex].
Thus, the solution to the equation [tex]\( x^{\frac{2}{3}} = 64 \)[/tex] is [tex]\( x = 512 \)[/tex].
From the given options, the correct answer is [tex]\( \boxed{512} \)[/tex].
