Question

use rational exponents to write as a single radical expression. assume t\\(\sqrt3{z} \cdot \sqrt4{z} \cdot \sqrt8{z^3}\\)\\(\sqrt3{z} \cdot \sqrt4{z} \cdot \sqrt8{z^3} = \square\\)(simplify your answer.)

Explanation:

Step1: Convert radicals to rational exponents

$\sqrt[3]{z} = z^{\frac{1}{3}}$, $\sqrt[4]{z} = z^{\frac{1}{4}}$, $\sqrt[8]{z^3} = z^{\frac{3}{8}}$

Step2: Add exponents (product rule)

$z^{\frac{1}{3} + \frac{1}{4} + \frac{3}{8}}$

Step3: Find common denominator (24)

$\frac{1}{3} = \frac{8}{24}$, $\frac{1}{4} = \frac{6}{24}$, $\frac{3}{8} = \frac{9}{24}$

Step4: Sum the fractions

$\frac{8 + 6 + 9}{24} = \frac{23}{24}$

Step5: Convert back to radical

$z^{\frac{23}{24}} = \sqrt[24]{z^{23}}$

Answer:

$\sqrt[24]{z^{23}}$