To solve the equation \( x^{2/3} = 64 \), we can eliminate the rational exponent by raising both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
- Raise both sides to \( \frac{3}{2} \):
\[ \left( x^{2/3} \right)^{3/2} = 64^{3/2} \]
- Simplifying the left side:
\[ x^{(2/3) \cdot (3/2)} = x^1 = x \]
- Now calculate \( 64^{3/2} \):
\[ 64^{3/2} = (64^{1/2})^3 \]
First find \( 64^{1/2} \):
\[ 64^{1/2} = 8 \]
Now raise 8 to the power of 3:
\[ 8^3 = 512 \]
- So, we have:
\[ x = 512 \]
The solution to the equation \( x^{2/3} = 64 \) is:
\[ \boxed{512} \]
1 answer