Question

\left(\frac{8^{-5}}{2^{-2}}\right)^{-4}\:equivalent\:expression

Answer 1

To simplify the expression \(\left(\frac{8^{-5}}{2^{-2}}\right)^{-4}\), we can follow these steps:

1. **Rewrite 8 in terms of base 2:**
\[
8 = 2^3
\]
Thus,
\[
8^{-5} = (2^3)^{-5} = 2^{-15}
\]

2. **Substitute back into the expression:**
\[
\frac{8^{-5}}{2^{-2}} = \frac{2^{-15}}{2^{-2}}
\]

3. **Simplify the fraction using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):**
\[
\frac{2^{-15}}{2^{-2}} = 2^{-15 - (-2)} = 2^{-15 + 2} = 2^{-13}
\]

4. **Now substitute this back into the original expression:**
\[
\left(2^{-13}\right)^{-4}
\]

5. **Apply the power of a power property \((a^m)^n = a^{m \cdot n}\):**
\[
\left(2^{-13}\right)^{-4} = 2^{-13 \cdot (-4)} = 2^{52}
\]

Thus, the equivalent expression is:
\[
\boxed{2^{52}}
\]