Question

simplify the expression given below.
$(81d^{24})^{-\frac{3}{4}}$
a. $-\frac{d^{18}}{27}$
b. $-27d^{18}$
c. $\frac{1}{27d^{18}}$
d. $\frac{27}{d^{18}}$

Explanation:

Step1: Rewrite negative exponent

$(81d^{24})^{-\frac{3}{4}} = \frac{1}{(81d^{24})^{\frac{3}{4}}}$

Step2: Split base using exponent rule

$\frac{1}{81^{\frac{3}{4}} \cdot (d^{24})^{\frac{3}{4}}}$

Step3: Simplify constant term

$81 = 3^4$, so $81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^{4 \times \frac{3}{4}} = 3^3 = 27$

Step4: Simplify variable term

$(d^{24})^{\frac{3}{4}} = d^{24 \times \frac{3}{4}} = d^{18}$

Step5: Combine results

$\frac{1}{27d^{18}}$

Answer:

C. $\frac{1}{27d^{18}}$