Factor (mn)^4-n^2

${(m n)}^{4} - n^{2}$

Step 1

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

${({(m n)}^{2})}^{2} - n^{2}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(m n)}^{2}$ and $b = n$ .

$({(m n)}^{2} + n) ({(m n)}^{2} - n)$

Step 3

Simplify.

Let $u = n$ . Substitute $u$ for all occurrences of $n$ .

$(m^{2} u^{2} + u) ({(m n)}^{2} - n)$

Factor $u$ out of $m^{2} u^{2} + u$ .

Factor $u$ out of $m^{2} u^{2}$ .

$(u (m^{2} u) + u) ({(m n)}^{2} - n)$

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

$(u (m^{2} u) + u^{1}) ({(m n)}^{2} - n)$

Factor $u$ out of $u^{1}$ .

$(u (m^{2} u) + u \cdot 1) ({(m n)}^{2} - n)$

Factor $u$ out of $u (m^{2} u) + u \cdot 1$ .

$u (m^{2} u + 1) ({(m n)}^{2} - n)$

Replace all occurrences of $u$ with $n$ .

$n (m^{2} n + 1) ({(m n)}^{2} - n)$

Let $u = n$ . Substitute $u$ for all occurrences of $n$ .

$n (m^{2} n + 1) (m^{2} u^{2} - u)$

Factor $u$ out of $m^{2} u^{2} - u$ .

Factor $u$ out of $m^{2} u^{2}$ .

$n (m^{2} n + 1) (u (m^{2} u) - u)$

Factor $u$ out of $- u$ .

$n (m^{2} n + 1) (u (m^{2} u) + u \cdot - 1)$

Factor $u$ out of $u (m^{2} u) + u \cdot - 1$ .

$n (m^{2} n + 1) (u (m^{2} u - 1))$

Factor.

Replace all occurrences of $u$ with $n$ .

$n (m^{2} n + 1) (n (m^{2} n - 1))$

Remove unnecessary parentheses.

$n (m^{2} n + 1) n (m^{2} n - 1)$

Combine exponents.

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

$n^{1} n (m^{2} n + 1) (m^{2} n - 1)$

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

$n^{1} n^{1} (m^{2} n + 1) (m^{2} n - 1)$

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

$n^{1 + 1} (m^{2} n + 1) (m^{2} n - 1)$

Add $1$ and $1$ .

$n^{2} (m^{2} n + 1) (m^{2} n - 1)$

Factor (mn)^4-n^2

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