Solution:
Given:
The expression;
[tex]2^5\times(\frac{1}{4})^7[/tex][tex]\begin{gathered} 2^5\times(\frac{1}{4})^7 \\ 2^5\times((\frac{1}{2})^2)^7 \\ Apply\text{ing the index law of powers,} \\ (x^a)^b=x^{ab}^{} \\ 2^5\times((\frac{1}{2})^2)^7=2^5\times(\frac{1}{2})^{14} \\ \\ \text{Thus,} \\ 2^5\times(\frac{1}{2})^{14}=2^5\times\frac{1^{14}}{2^{14}} \\ 2^5\times\frac{1^{14}}{2^{14}}=2^5\times\frac{1^{}}{2^{14}} \\ \\ \text{Applying the law of negative exponent,} \\ x^{-a}=\frac{1}{x^a} \\ 2^5\times\frac{1^{}}{2^{14}}=2^5\times2^{-14} \\ \\ \text{Apply ing the product law of indices,} \\ x^a\times x^b=x^{a+b} \\ \\ \text{Hence, } \\ 2^5\times2^{-14}=2^{5+(-14)} \\ 2^{5+(-14)}=2^{5-14} \\ =2^{-9} \end{gathered}[/tex]
Therefore, the correct answer is OPTION C.
