Indices

Concept:Simplify using laws of indices (base 2 for first term, base 5 for second) to reduce the expression to a constant.Explanation:Rewrite the first fraction in base 2: $16 = 2^4$, $32 = 2^5$, $4 = 2^2$.Numerator: $16(32)^x = 2^4 \cdot 2^{5x} = 2^{5x+4}$; $2^{3x-2} \cdot 4^{x+1} = 2^{3x-2} \cdot 2^{2x+2} = 2^{5x}$.So numerator becomes $2^{5x+4} - 2^{5x} = 2^{5x}(16-1) = 15 \cdot 2^{5x}$.Denominator: $15(2)^{x-1}(16)^x = 15 \cdot 2^{x-1} \cdot 2^{4x} = 15 \cdot 2^{5x-1}$.Thus first fraction = $\frac{15 \cdot 2^{5x}}{15 \cdot 2^{5x-1}} = 2^{1} = 2$.Simplify the second fraction: numerator $5(5)^{x-1} = 5^{x}$.Denominator $\sqrt{5^{2x}} = 5^{x}$.Thus second fraction = $\frac{5^x}{5^x} = 1$.Therefore, the entire expression = $2 - 1 = 1$.Answer:$1$