Simplify the Radical Expression ( fourth root of p^7q^10)/( fourth root of p^2q^2)

$\frac{\sqrt[4]{p^{7} q^{10}}}{\sqrt[4]{p^{2} q^{2}}}$

Step 1

Combine $\sqrt[4]{p^{7} q^{10}}$ and $\sqrt[4]{p^{2} q^{2}}$ into a single radical.

$\sqrt[4]{\frac{p^{7} q^{10}}{p^{2} q^{2}}}$

Step 2

Reduce the expression $\frac{p^{7} q^{10}}{p^{2} q^{2}}$ by cancelling the common factors.

Factor $p^{2}$ out of $p^{7} q^{10}$ .

$\sqrt[4]{\frac{p^{2} (p^{5} q^{10})}{p^{2} q^{2}}}$

Factor $p^{2}$ out of $p^{2} q^{2}$ .

$\sqrt[4]{\frac{p^{2} (p^{5} q^{10})}{p^{2} (q^{2})}}$

Cancel the common factor.

$\sqrt[4]{\frac{p^{2} (p^{5} q^{10})}{p^{2} q^{2}}}$

Rewrite the expression.

$\sqrt[4]{\frac{p^{5} q^{10}}{q^{2}}}$

Step 3

Cancel the common factor of $q^{10}$ and $q^{2}$ .

Factor $q^{2}$ out of $p^{5} q^{10}$ .

$\sqrt[4]{\frac{q^{2} (p^{5} q^{8})}{q^{2}}}$

Cancel the common factors.

Multiply by $1$ .

$\sqrt[4]{\frac{q^{2} (p^{5} q^{8})}{q^{2} \cdot 1}}$

Cancel the common factor.

$\sqrt[4]{\frac{q^{2} (p^{5} q^{8})}{q^{2} \cdot 1}}$

Rewrite the expression.

$\sqrt[4]{\frac{p^{5} q^{8}}{1}}$

Divide $p^{5} q^{8}$ by $1$ .

$\sqrt[4]{p^{5} q^{8}}$

Step 4

Rewrite $p^{5} q^{8}$ as ${(p q^{2})}^{4} p$ .

Factor out $p^{4}$ .

$\sqrt[4]{p^{4} p q^{8}}$

Rewrite $q^{8}$ as ${(q^{2})}^{4}$ .

$\sqrt[4]{p^{4} p {(q^{2})}^{4}}$

Move $p$ .

$\sqrt[4]{p^{4} {(q^{2})}^{4} p}$

Rewrite $p^{4} {(q^{2})}^{4}$ as ${(p q^{2})}^{4}$ .

$\sqrt[4]{{(p q^{2})}^{4} p}$

Step 5

Pull terms out from under the radical.

$| p q^{2} | \sqrt[4]{p}$

Step 6

Remove non-negative terms from the absolute value.

$q^{2} | p | \sqrt[4]{p}$