Question

which expression is equivalent to $32\sqrt3{18y} \div 8\sqrt3{3y}$, if $y \
eq 0$?

a. $12\sqrt3{2y^2}$

b. $4\sqrt3{6}$

c. $4\sqrt3{15y}$

d. $4\sqrt3{6y}$

Explanation:

Step1: Divide the coefficients

Divide the coefficients 32 and 8. So, \( \frac{32}{8} = 4 \).

Step2: Divide the cube - root terms

For the cube - root terms, we have \( \frac{\sqrt[3]{18y}}{\sqrt[3]{3y}}=\sqrt[3]{\frac{18y}{3y}} \) (using the property \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \) for \( b
eq0 \)).
Simplify \( \frac{18y}{3y} \), the \( y \) terms cancel out (since \( y
eq0 \)) and \( \frac{18}{3}=6 \). So \( \sqrt[3]{\frac{18y}{3y}}=\sqrt[3]{6} \).

Step3: Multiply the results from Step1 and Step2

Multiply the coefficient result (4) with the cube - root result (\( \sqrt[3]{6} \)). So we get \( 4\sqrt[3]{6} \).

Answer:

B. \( 4\sqrt[3]{6} \)