Basic Math Examples

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

Step 1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{{(m - n)}^{2}}{{(m - n)}^{2}}$ .

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

Step 2

Combine $- 1$ and $\frac{{(m - n)}^{2}}{{(m - n)}^{2}}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Rewrite $\frac{m + n^{2} - {(m - n)}^{2}}{{(m - n)}^{2}}$ in a factored form.

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Step 4.1

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

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Step 4.1.1

Subtract $n^{2}$ from $n^{2}$ .

$\frac{u - u^{2} + 2 n u + 0}{{(m - n)}^{2}}$

Step 4.1.2

Add $u - u^{2} + 2 n u$ and $0$ .

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

Step 4.2

Factor $- u$ out of $u - u^{2} + 2 n u$ .

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Step 4.2.1

Reorder the expression.

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Step 4.2.1.1

Move $u$ .

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

Step 4.2.1.2

Reorder $- u^{2}$ and $2 n u$ .

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

Step 4.2.2

Factor $- u$ out of $2 n u$ .

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

Step 4.2.3

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

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

Step 4.2.4

Rewrite $u$ as $1 u$ .

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

Step 4.2.5

Factor $- u$ out of $1 u$ .

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

Step 4.2.6

Factor $- u$ out of $- u (- 2 n) - u (u)$ .

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

Step 4.2.7

Factor $- u$ out of $- u (- 2 n + u) - u (- 1)$ .

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

Step 4.3

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

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

Step 5

Factor.

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Step 5.1

Factor $- 1$ out of $- 2 n + m - 1$ .

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Step 5.1.1

Reorder $- 2 n$ and $m$ .

$\frac{- m (m - 2 n - 1)}{{(m - n)}^{2}}$

Step 5.1.2

Factor $- 1$ out of $m$ .

$\frac{- m (- 1 (- m) - 2 n - 1)}{{(m - n)}^{2}}$

Step 5.1.3

Factor $- 1$ out of $- 2 n$ .

$\frac{- m (- 1 (- m) - (2 n) - 1)}{{(m - n)}^{2}}$

Step 5.1.4

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- m (- 1 (- m) - (2 n) - 1 (1))}{{(m - n)}^{2}}$

Step 5.1.5

Factor $- 1$ out of $- 1 (- m) - (2 n)$ .

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

Step 5.1.6

Factor $- 1$ out of $- 1 (- m + 2 n) - 1 (1)$ .

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

Step 5.1.7

Rewrite $- 1 (- m + 2 n + 1)$ as $- (- m + 2 n + 1)$ .

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

Combine exponents.

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Step 6.1

Factor out negative.

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

Step 6.2

Multiply $- 1$ by $- 1$ .

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

Step 6.3

Multiply $m$ by $1$ .

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

Step 7

Factor.

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Step 7.1

Factor $- 1$ out of $- m + 2 n + 1$ .

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Step 7.1.1

Factor $- 1$ out of $- m$ .

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

Step 7.1.2

Factor $- 1$ out of $2 n$ .

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

Step 7.1.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{m (- (m) - (- 2 n) - 1 (- 1))}{{(m - n)}^{2}}$

Step 7.1.4

Factor $- 1$ out of $- (m) - (- 2 n)$ .

$\frac{m (- (m - 2 n) - 1 (- 1))}{{(m - n)}^{2}}$

Step 7.1.5

Factor $- 1$ out of $- (m - 2 n) - 1 (- 1)$ .

$\frac{m (- (m - 2 n - 1))}{{(m - n)}^{2}}$

Step 7.2

Remove unnecessary parentheses.

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

Step 8

Factor out negative.

$\frac{- (m (m - 2 n - 1))}{{(m - n)}^{2}}$

Step 9

Remove unnecessary parentheses.

$\frac{- m (m - 2 n - 1)}{{(m - n)}^{2}}$