Question

Write problem as a single radical using the smallest possible root. 16

Answer 1

The radical expression is given to be:

$\sqrt{n^5}\sqrt[3]{n^4}$

Recall the rule:

$\sqrt[n]{x}=x^{\frac{1}{n}}$

Therefore, the expression becomes:

$\sqrt{n^5}\sqrt[3]{n^4}=n^{\frac{5}{2}}\cdot n^{\frac{4}{3}}$

Recall the rule:

$x^a\cdot x^b=x^{a+b}$

Therefore, we have:

$n^5\cdot n^{\frac{4}{3}}=n^{\frac{5}{2}+\frac{4}{3}}$

Since:

$\frac{5}{2}+\frac{4}{3}=\frac{23}{6}$

Therefore, the expression becomes:

$n^{\frac{5}{2}+\frac{4}{3}}=n^{\frac{23}{6}}$

Therefore, we can rewrite the radical to be:

$\Rightarrow\sqrt[6]{n^{23}}$