Basic Math Examples

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

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

Step 1

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

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

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

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

$- 1$ and

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

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

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

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

Step 4

Rewrite

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

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

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Step 4.1

Let

u = m

$u = m$ . Substitute

u

$u$ for all occurrences of

m

$m$ .

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Step 4.1.1

Subtract

n^{2}

$n^{2}$ from

n^{2}

$n^{2}$ .

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

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

Step 4.1.2

Add

u - u^{2} + 2 n u

$u - u^{2} + 2 n u$ and

0

$0$ .

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

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

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

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

Step 4.2

Factor

- u

$- u$ out of

u - u^{2} + 2 n u

$u - u^{2} + 2 n u$ .

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Step 4.2.1

Reorder the expression.

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Step 4.2.1.1

Move

u

$u$ .

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

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

Step 4.2.1.2

Reorder

- u^{2}

$- u^{2}$ and

2 n u

$2 n u$ .

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

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

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

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

Step 4.2.2

Factor

- u

$- u$ out of

2 n u

$2 n u$ .

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

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

Step 4.2.3

Factor

- u

$- u$ out of

- u^{2}

$- u^{2}$ .

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

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

Step 4.2.4

Rewrite

u

$u$ as

1 u

$1 u$ .

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

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

Step 4.2.5

Factor

- u

$- u$ out of

1 u

$1 u$ .

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

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

Step 4.2.6

Factor

- u

$- u$ out of

- u (- 2 n) - u (u)

$- u (- 2 n) - u (u)$ .

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

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

Step 4.2.7

Factor

- u

$- u$ out of

- u (- 2 n + u) - u (- 1)

$- u (- 2 n + u) - u (- 1)$ .

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

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

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

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

Step 4.3

Replace all occurrences of

u

$u$ with

m

$m$ .

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

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

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

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

Step 5

Factor.

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Step 5.1

Factor

- 1

$- 1$ out of

- 2 n + m - 1

$- 2 n + m - 1$ .

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Step 5.1.1

Reorder

- 2 n

$- 2 n$ and

m

$m$ .

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

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

Step 5.1.2

Factor

- 1

$- 1$ out of

m

$m$ .

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

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

Step 5.1.3

Factor

- 1

$- 1$ out of

- 2 n

$- 2 n$ .

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

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

Step 5.1.4

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 5.1.5

Factor

- 1

$- 1$ out of

- 1 (- m) - (2 n)

$- 1 (- m) - (2 n)$ .

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

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

Step 5.1.6

Factor

- 1

$- 1$ out of

- 1 (- m + 2 n) - 1 (1)

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

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

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

Step 5.1.7

Rewrite

- 1 (- m + 2 n + 1)

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

- (- m + 2 n + 1)

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

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

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

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

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

Step 5.2

Remove unnecessary parentheses.

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

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

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

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

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}}

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

Step 6.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 6.3

Multiply

m

$m$ by

1

$1$ .

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

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

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

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

Step 7

Factor.

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Step 7.1

Factor

- 1

$- 1$ out of

- m + 2 n + 1

$- m + 2 n + 1$ .

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Step 7.1.1

Factor

- 1

$- 1$ out of

- m

$- m$ .

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

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

Step 7.1.2

Factor

- 1

$- 1$ out of

2 n

$2 n$ .

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

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

Step 7.1.3

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 7.1.4

Factor

- 1

$- 1$ out of

- (m) - (- 2 n)

$- (m) - (- 2 n)$ .

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

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

Step 7.1.5

Factor

- 1

$- 1$ out of

- (m - 2 n) - 1 (- 1)

$- (m - 2 n) - 1 (- 1)$ .

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

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

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

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

Step 7.2

Remove unnecessary parentheses.

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

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

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

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

Step 8

Factor out negative.

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

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

Step 9

Remove unnecessary parentheses.

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

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