Simplify 4 square root of n^2+ square root of m^2n- square root of 4n^2- square root of mn^2

$4 \sqrt{n^{2}} + \sqrt{m^{2} n} - \sqrt{4 n^{2}} - \sqrt{m n^{2}}$

Step 1

Simplify each term.

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

$4 n + \sqrt{m^{2} n} - \sqrt{4 n^{2}} - \sqrt{m n^{2}}$

Pull terms out from under the radical.

$4 n + m \sqrt{n} - \sqrt{4 n^{2}} - \sqrt{m n^{2}}$

Rewrite $4 n^{2}$ as ${(2 n)}^{2}$ .

$4 n + m \sqrt{n} - \sqrt{{(2 n)}^{2}} - \sqrt{m n^{2}}$

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

$4 n + m \sqrt{n} - (2 n) - \sqrt{m n^{2}}$

Multiply $2$ by $- 1$ .

$4 n + m \sqrt{n} - 2 n - \sqrt{m n^{2}}$

Reorder $m$ and $n^{2}$ .

$4 n + m \sqrt{n} - 2 n - \sqrt{n^{2} m}$

Pull terms out from under the radical.

$4 n + m \sqrt{n} - 2 n - n \sqrt{m}$

Step 2

Subtract $2 n$ from $4 n$ .

$m \sqrt{n} + 2 n - n \sqrt{m}$