Question

1. $(36a^{2})^{-\frac{3}{2}}$

$(36)^{-\frac{3}{2}}(a^{2})^{-\frac{3}{2}}$
$a$

Explanation:

Step1: Apply exponent product rule

$(ab)^n = a^n b^n$
So, $(36a^2)^{-\frac{3}{2}} = 36^{-\frac{3}{2}} \cdot (a^2)^{-\frac{3}{2}}$

Step2: Simplify constant term

Rewrite negative exponent: $36^{-\frac{3}{2}} = \frac{1}{36^{\frac{3}{2}}}$
$36^{\frac{3}{2}} = (36^{\frac{1}{2}})^3 = 6^3 = 216$
Thus, $36^{-\frac{3}{2}} = \frac{1}{216}$

Step3: Simplify variable term

Apply power rule: $(a^m)^n = a^{m \cdot n}$
$(a^2)^{-\frac{3}{2}} = a^{2 \cdot (-\frac{3}{2})} = a^{-3} = \frac{1}{a^3}$

Step4: Combine results

Multiply the simplified constant and variable terms
$\frac{1}{216} \cdot \frac{1}{a^3}$

Answer:

$\frac{1}{216a^3}$