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

To solve the equation \( 5^{-6} \div 5^{p} = 125^{\circ} \), we first recognize that \( 125 \) can be rewritten as \( 5^3 \). Therefore, \( 125^{\circ} = (5^3)^{\circ} \). Next, we know \( a^{b^{c}} = a^{bc} \), leading to \( (5^3)^{2} = 5^{3 \cdot 2} = 5^6 \). Hence, the equation becomes: \[ 5^{-6} \div 5^{p} = 5^6 \] Using the property of exponents that \( a^m \div a^n = a^{m-n} \), we can simplify the left side: \[ 5^{-6 - p} = 5^6 \] Since the bases are the same, we can set the exponents equal to each other: \[ -6 - p = 6 \] To solve for \( p \), we rearrange the equation: \[ -p = 6 + 6 \implies -p = 12 \implies p = -12 \] Now, we need to find \( \frac{1}{2^{p}} \): \[ \frac{1}{2^{p}} = \frac{1}{2^{-12}} = 2^{12} \] Now calculating \( 2^{12} \): \[ 2^{12} = 4096 \] Therefore, the value of \( \frac{1}{2^{p}} \) is \( \boxed{4096} \).