Question

Simplify.
64^–5/6

Answer 1

To simplify \(64^{-5/6}\), we can follow these steps:

1. **Rewrite the negative exponent**:
\[
64^{-5/6} = \frac{1}{64^{5/6}}
\]

2. **Rewrite \(64\) as a power of \(4\)**:
\[
64 = 4^3 \quad \text{(since \(4^3 = 64\))}
\]
Thus,
\[
64^{5/6} = (4^3)^{5/6}
\]

3. **Use the exponent multiplication rule**:
\[
(a^m)^n = a^{m \cdot n}
\]
So,
\[
(4^3)^{5/6} = 4^{3 \cdot (5/6)} = 4^{15/6} = 4^{5/2}
\]

4. **Convert \(4^{5/2}\) into a more manageable form**:
\[
4^{5/2} = (4^{1/2})^5 = (2)^5 = 32 \quad \text{(as \(4^{1/2} = 2\))}
\]

5. **Combine it back into the fraction**:
\[
64^{-5/6} = \frac{1}{64^{5/6}} = \frac{1}{4^{5/2}} = \frac{1}{32}
\]

Thus, the simplified result is:
\[
\boxed{\frac{1}{32}}
\]