Simplify the Radical Expression ((4m^2n)^2)/(2m^5n)

$\frac{{(4 m^{2} n)}^{2}}{2 m^{5} n}$

Step 1

Simplify the numerator.

Apply the product rule to $4 m^{2} n$ .

$\frac{{(4 m^{2})}^{2} n^{2}}{2 m^{5} n}$

Apply the product rule to $4 m^{2}$ .

$\frac{4^{2} {(m^{2})}^{2} n^{2}}{2 m^{5} n}$

Raise $4$ to the power of $2$ .

$\frac{16 {(m^{2})}^{2} n^{2}}{2 m^{5} n}$

Multiply the exponents in ${(m^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{16 m^{2 \cdot 2} n^{2}}{2 m^{5} n}$

Multiply $2$ by $2$ .

$\frac{16 m^{4} n^{2}}{2 m^{5} n}$

$\frac{16 m^{4} n^{2}}{2 m^{5} n}$

$\frac{16 m^{4} n^{2}}{2 m^{5} n}$

Step 2

Cancel the common factor of $16$ and $2$ .

Factor $2$ out of $16 m^{4} n^{2}$ .

$\frac{2 (8 m^{4} n^{2})}{2 m^{5} n}$

Cancel the common factors.

Factor $2$ out of $2 m^{5} n$ .

$\frac{2 (8 m^{4} n^{2})}{2 (m^{5} n)}$

Cancel the common factor.

$\frac{2 (8 m^{4} n^{2})}{2 (m^{5} n)}$

Rewrite the expression.

$\frac{8 m^{4} n^{2}}{m^{5} n}$

$\frac{8 m^{4} n^{2}}{m^{5} n}$

$\frac{8 m^{4} n^{2}}{m^{5} n}$

Step 3

Cancel the common factor of $m^{4}$ and $m^{5}$ .

Factor $m^{4}$ out of $8 m^{4} n^{2}$ .

$\frac{m^{4} (8 n^{2})}{m^{5} n}$

Cancel the common factors.

Factor $m^{4}$ out of $m^{5} n$ .

$\frac{m^{4} (8 n^{2})}{m^{4} (m n)}$

Cancel the common factor.

$\frac{m^{4} (8 n^{2})}{m^{4} (m n)}$

Rewrite the expression.

$\frac{8 n^{2}}{m n}$

$\frac{8 n^{2}}{m n}$

$\frac{8 n^{2}}{m n}$

Step 4

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

Factor $n$ out of $8 n^{2}$ .

$\frac{n (8 n)}{m n}$

Cancel the common factors.

Factor $n$ out of $m n$ .

$\frac{n (8 n)}{n m}$

Cancel the common factor.

$\frac{n (8 n)}{n m}$

Rewrite the expression.

$\frac{8 n}{m}$

$\frac{8 n}{m}$

$\frac{8 n}{m}$

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