To solve the equation \( x^6 = 64 \), we first rewrite \( 64 \) as a power of \( 2 \):
\[ 64 = 2^6 \]
Now, we can rewrite the equation:
\[ x^6 = 2^6 \]
Taking the sixth root of both sides gives:
\[ x = 2 \]
Since we're dealing with an even exponent, there are actually two solutions (one positive and one negative):
\[ x = 2 \quad \text{or} \quad x = -2 \]
Thus, the values of \( x \) that satisfy the equation \( x^6 = 64 \) are:
\[ \boxed{2 \text{ and } -2} \]