Simplify the Radical Expression eighth root of (n^18p^7)/(n^2p^-1)

$\sqrt[8]{\frac{n^{18} p^{7}}{n^{2} p^{- 1}}}$

Step 1

Reduce the expression $\frac{n^{18} p^{7}}{n^{2} p^{- 1}}$ by cancelling the common factors.

Factor $n^{2}$ out of $n^{18} p^{7}$ .

$\sqrt[8]{\frac{n^{2} (n^{16} p^{7})}{n^{2} p^{- 1}}}$

Factor $n^{2}$ out of $n^{2} p^{- 1}$ .

$\sqrt[8]{\frac{n^{2} (n^{16} p^{7})}{n^{2} (p^{- 1})}}$

Cancel the common factor.

$\sqrt[8]{\frac{n^{2} (n^{16} p^{7})}{n^{2} p^{- 1}}}$

Rewrite the expression.

$\sqrt[8]{\frac{n^{16} p^{7}}{p^{- 1}}}$

Step 2

Move $p^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt[8]{n^{16} p^{7} p}$

Step 3

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

Move $p$ .

$\sqrt[8]{n^{16} (p \cdot p^{7})}$

Multiply $p$ by $p^{7}$ .

Raise $p$ to the power of $1$ .

$\sqrt[8]{n^{16} (p^{1} p^{7})}$

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

$\sqrt[8]{n^{16} p^{1 + 7}}$

Add $1$ and $7$ .

$\sqrt[8]{n^{16} p^{8}}$

Step 4

Rewrite $n^{16} p^{8}$ as ${(n^{2} p)}^{8}$ .

$\sqrt[8]{{(n^{2} p)}^{8}}$

Step 5

Pull terms out from under the radical.

$| n^{2} p |$

Step 6

Remove non-negative terms from the absolute value.

$n^{2} | p |$