Question

use the properties of rational exponents to simplify $\frac{2^{2/3} cdot 16^{2/3}}{4^{2/3}}$.
$\bigcirc$ $2^{10/3}$
$\bigcirc$ $2^{8/3}$
$\bigcirc$ $2^{4/3}$
$\bigcirc$ $4$

Explanation:

Step1: Rewrite bases as powers of 2

$16=2^4$, $4=2^2$, so:
$$\frac{2^{2/3} \cdot (2^4)^{2/3}}{(2^2)^{2/3}}$$

Step2: Simplify exponents via power rule

Power rule: $(a^m)^n=a^{m \cdot n}$:
$$\frac{2^{2/3} \cdot 2^{8/3}}{2^{4/3}}$$

Step3: Combine numerator exponents

Product rule: $a^m \cdot a^n=a^{m+n}$:
$$\frac{2^{\frac{2}{3}+\frac{8}{3}}}{2^{4/3}}=\frac{2^{10/3}}{2^{4/3}}$$

Step4: Subtract exponents (quotient rule)

Quotient rule: $\frac{a^m}{a^n}=a^{m-n}$:
$$2^{\frac{10}{3}-\frac{4}{3}}=2^{6/3}=2^2$$

Step5: Simplify the final power

$$2^2=4$$

Answer:

4