Question

which expression is equivalent to $40x^{2}\sqrt{x^{12}} \div 10x^{2}\sqrt{29x^{8}}$, if $x \
eq 0$?\
\
a. $\frac{4x^{2}\sqrt{29}}{\sqrt{29}}$\
\
b. $\frac{30x^{2}\sqrt{29}}{29}$\
\
c. $4x^{2}\sqrt{29}$\
\
d. $\frac{4x^{2}\sqrt{29}}{29}$

Explanation:

Step1: Rewrite division as fraction

$$\frac{40x^2\sqrt{x^{12}}}{10x^2\sqrt{29x^8}}$$

Step2: Simplify coefficients and $x^2$

$$\frac{40}{10} \cdot \frac{x^2}{x^2} \cdot \frac{\sqrt{x^{12}}}{\sqrt{29x^8}} = 4 \cdot \frac{\sqrt{x^{12}}}{\sqrt{29x^8}}$$

Step3: Combine square roots

$$4 \cdot \sqrt{\frac{x^{12}}{29x^8}} = 4 \cdot \sqrt{\frac{x^4}{29}}$$

Step4: Simplify $\sqrt{x^4}$ and rationalize

$$4 \cdot \frac{x^2}{\sqrt{29}} = 4 \cdot \frac{x^2\sqrt{29}}{29} = \frac{4x^2\sqrt{29}}{29}$$

Answer:

D. $\frac{4x^2\sqrt{29}}{29}$