Question

1. simplify the expression below.

$(2x)^{\frac{1}{3}} cdot (2x)^{\frac{4}{3}}$
a. $2xsqrt3{4x^2}$
b. $2xsqrt3{2x}$
c. $4xsqrt3{4x^2}$
d. $4xsqrt3{2x}$

Explanation:

Step1: Apply exponent product rule

When multiplying terms with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$.
$$(2x)^{\frac{1}{3} + \frac{4}{3}} = (2x)^{\frac{5}{3}}$$

Step2: Split the exponent

Rewrite the exponent as a sum of integer and fraction: $\frac{5}{3} = 1 + \frac{2}{3}$.
$$(2x)^{1 + \frac{2}{3}} = (2x)^1 \cdot (2x)^{\frac{2}{3}}$$

Step3: Expand and rewrite in radical form

Convert the fractional exponent to a cube root: $a^{\frac{2}{3}} = \sqrt[3]{a^2}$.
$$2x \cdot \sqrt[3]{(2x)^2} = 2x \cdot \sqrt[3]{4x^2}$$

Step4: Rearrange to match option format

$$2x\sqrt[3]{4x^2}$$

Answer:

A. $2x\sqrt[3]{4x^2}$