Given that $$ 5^{-6} \div 5^{p}=125^{\circ} $$, find the value of $$ \frac{1}{2^{p}} $$

Question

UpStudy AI · Accepted Answer

1. Express the division with exponents using the exponent rule:
   $$
   5^{-6} \div 5^p = 5^{-6-p}.
   $$

2. Recognize that $$125 = 5^3$$. Although the original problem writes $$125^\circ$$, we interpret it as $$125$$ based on exponent matching.

3. Set the exponents equal:
   $$
   5^{-6-p} = 5^3 \quad \Longrightarrow \quad -6-p = 3.
   $$

4. Solve for $$p$$:
   $$
   -p = 3 + 6 = 9 \quad \Longrightarrow \quad p = -9.
   $$

5. Substitute $$p = -9$$ into $$\frac{1}{2^p}$$:
   $$
   \frac{1}{2^p} = \frac{1}{2^{-9}} = 2^9.
   $$

6. Calculate $$2^9$$:
   $$
   2^9 = 512.
   $$

Thus, the value of $$\frac{1}{2^p}$$ is $$\boxed{512}$$.
$$ \frac{1}{2^p} = 512 $$

Given that , find the value of