Question

use rational exponents to write as a single radical expression. assume that all variables represent positive real numbers
\\(\sqrt5{z} \cdot \sqrt4{z} \cdot \sqrt8{z^3}\\)

\\(\sqrt5{z} \cdot \sqrt4{z} \cdot \sqrt8{z^3} = \square\\)
(simplify your answer)

Explanation:

Step1: Convert radicals to rational exponents

Recall that $\sqrt[n]{a}=a^{\frac{1}{n}}$ and $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. So we can rewrite each radical:
$\sqrt[5]{z}=z^{\frac{1}{5}}$, $\sqrt[4]{z}=z^{\frac{1}{4}}$, $\sqrt[8]{z^3}=z^{\frac{3}{8}}$

Step2: Multiply the expressions using exponent rule $a^m \cdot a^n = a^{m + n}$

Now we multiply them together: $z^{\frac{1}{5}} \cdot z^{\frac{1}{4}} \cdot z^{\frac{3}{8}}$
To add the exponents, we need a common denominator. The least common denominator of 5, 4, and 8 is 40.
Convert each fraction:
$\frac{1}{5}=\frac{8}{40}$, $\frac{1}{4}=\frac{10}{40}$, $\frac{3}{8}=\frac{15}{40}$
Now add the exponents: $\frac{8}{40}+\frac{10}{40}+\frac{15}{40}=\frac{8 + 10 + 15}{40}=\frac{33}{40}$
So the expression becomes $z^{\frac{33}{40}}$

Step3: Convert back to radical form

Since $a^{\frac{m}{n}}=\sqrt[n]{a^m}$, we can rewrite $z^{\frac{33}{40}}$ as $\sqrt[40]{z^{33}}$

Answer:

$\sqrt[40]{z^{33}}$