Question

6 multiple answer 1 point choose all expressions that are equivalent to: ((54x^{6}y^{4})^{\frac{1}{3}}) (27x^{2}y^{\frac{4}{3}}) (27x^{2}ysqrt3{y}) no equivalent expressions (3x^{2}ysqrt3{2y}) (54^{\frac{1}{3}}x^{2}y^{\frac{4}{3}})

Explanation:

Step1: Apply exponent product rule

$$(54x^6y^4)^{\frac{1}{3}} = 54^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}} \cdot (y^4)^{\frac{1}{3}}$$

Step2: Simplify variable exponents

$$(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2$$
$$(y^4)^{\frac{1}{3}} = y^{4 \cdot \frac{1}{3}} = y^{\frac{4}{3}}$$

Step3: Rewrite $54^{\frac{1}{3}}$ and $y^{\frac{4}{3}}$

$$54^{\frac{1}{3}} = (27 \cdot 2)^{\frac{1}{3}} = 27^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} = 3 \cdot 2^{\frac{1}{3}}$$
$$y^{\frac{4}{3}} = y^{1+\frac{1}{3}} = y \cdot y^{\frac{1}{3}} = y\sqrt[3]{y}$$

Step4: Combine simplified terms

$$3 \cdot 2^{\frac{1}{3}} \cdot x^2 \cdot y \cdot y^{\frac{1}{3}} = 3x^2y\sqrt[3]{2y}$$

Step5: Match to options

Also, $54^{\frac{1}{3}}x^2y^{\frac{4}{3}}$ is the direct expanded form from Step1.

Answer:

$3x^2 y\sqrt[3]{2y}$, $54^{\frac{1}{3}} x^2 y^{\frac{4}{3}}$