Simplify fourth root of 1/(p^5*p^7)

$\sqrt[4]{\frac{1}{p^{5} \cdot p^{7}}}$

Step 1

Multiply $p^{5}$ by $p^{7}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[4]{\frac{1}{p^{5 + 7}}}$

Add $5$ and $7$ .

$\sqrt[4]{\frac{1}{p^{12}}}$

Step 2

Rewrite $1$ as $1^{4}$ .

$\sqrt[4]{\frac{1^{4}}{p^{12}}}$

Step 3

Rewrite $p^{12}$ as ${(p^{3})}^{4}$ .

$\sqrt[4]{\frac{1^{4}}{{(p^{3})}^{4}}}$

Step 4

Rewrite $\frac{1^{4}}{{(p^{3})}^{4}}$ as ${(\frac{1}{p^{3}})}^{4}$ .

$\sqrt[4]{{(\frac{1}{p^{3}})}^{4}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{p^{3}}$