Question

1 multiple choice 1 point
which of the following is equivalent to the expression $4\sqrt3{ab^9}$?
$\frac{4}{a^{\frac{1}{3}}b^3}$
$4a^3b^{\frac{1}{3}}$
$4a^{\frac{1}{3}}b^3$
$4a^3b^{27}$

Explanation:

Step1: Convert root to rational exponent

Recall that $\sqrt[n]{x}=x^{\frac{1}{n}}$, so $\sqrt[3]{ab^9}=(ab^9)^{\frac{1}{3}}$.

Step2: Apply exponent distribution

Use $(xy)^n=x^n y^n$: $(ab^9)^{\frac{1}{3}}=a^{\frac{1}{3}}(b^9)^{\frac{1}{3}}$.

Step3: Simplify $b$ exponent

Use $(x^m)^n=x^{mn}$: $(b^9)^{\frac{1}{3}}=b^{9\times\frac{1}{3}}=b^3$.

Step4: Combine with coefficient

Multiply by the constant 4: $4a^{\frac{1}{3}}b^3$.

Answer:

$4a^{\frac{1}{3}}b^3$ (the third option)