Basic Math Examples

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

Step 1

To write $\frac{m + n}{m - n}$ as a fraction with a common denominator, multiply by $\frac{m + n}{m + n}$ .

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

Step 2

To write $- \frac{m - n}{m + n}$ as a fraction with a common denominator, multiply by $\frac{m - n}{m - n}$ .

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

Step 3

Write each expression with a common denominator of $(m - n) (m + n)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{m + n}{m - n}$ by $\frac{m + n}{m + n}$ .

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

Step 3.2

Multiply $\frac{m - n}{m + n}$ by $\frac{m - n}{m - n}$ .

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

Step 3.3

Reorder the factors of $(m - n) (m + n)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Simplify.

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Step 5.7.1

Add $m$ and $m$ .

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

Step 5.7.2

Subtract $n$ from $n$ .

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

Step 5.7.3

Add $2 m$ and $0$ .

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

Step 5.7.4

Apply the distributive property.

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

Step 5.7.5

Multiply $- - n$ .

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Step 5.7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.7.5.2

Multiply $n$ by $1$ .

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

Step 5.7.6

Subtract $m$ from $m$ .

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

Step 5.7.7

Add $n$ and $0$ .

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

Step 5.7.8

Add $n$ and $n$ .

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

Step 5.7.9

Multiply $2$ by $2$ .

$(\frac{m}{n} - \frac{n}{m}) \cdot \frac{4 m n}{(m + n) (m - n)}$

Step 6

Multiply $\frac{m}{n} - \frac{n}{m}$ by $\frac{4 m n}{(m + n) (m - n)}$ .

$\frac{(\frac{m}{n} - \frac{n}{m}) (4 m n)}{(m + n) (m - n)}$

Step 7

Simplify the numerator.

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Step 7.1

To write $\frac{m}{n}$ as a fraction with a common denominator, multiply by $\frac{m}{m}$ .

$\frac{(\frac{m}{n} \cdot \frac{m}{m} - \frac{n}{m}) \cdot 4 m n}{(m + n) (m - n)}$

Step 7.2

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

$\frac{(\frac{m}{n} \cdot \frac{m}{m} - \frac{n}{m} \cdot \frac{n}{n}) \cdot 4 m n}{(m + n) (m - n)}$

Step 7.3

Write each expression with a common denominator of $n m$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.1

Multiply $\frac{m}{n}$ by $\frac{m}{m}$ .

$\frac{(\frac{m \cdot m}{n m} - \frac{n}{m} \cdot \frac{n}{n}) \cdot 4 m n}{(m + n) (m - n)}$

Step 7.3.2

Multiply $\frac{n}{m}$ by $\frac{n}{n}$ .

$\frac{(\frac{m \cdot m}{n m} - \frac{n \cdot n}{m n}) \cdot 4 m n}{(m + n) (m - n)}$

Step 7.3.3

Reorder the factors of $n m$ .

$\frac{(\frac{m \cdot m}{m n} - \frac{n \cdot n}{m n}) \cdot 4 m n}{(m + n) (m - n)}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{\frac{m \cdot m - n \cdot n}{m n} \cdot 4 m n}{(m + n) (m - n)}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

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

$\frac{\frac{m^{1} m - n \cdot n}{m n} \cdot 4 m n}{(m + n) (m - n)}$

Step 7.5.2

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

$\frac{\frac{m^{1} m^{1} - n \cdot n}{m n} \cdot 4 m n}{(m + n) (m - n)}$

Step 7.5.3

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

$\frac{\frac{m^{1 + 1} - n \cdot n}{m n} \cdot 4 m n}{(m + n) (m - n)}$

Step 7.5.4

Add $1$ and $1$ .

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

Step 7.5.5

Rewrite $n \cdot n$ as $n^{2}$ .

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

Step 7.5.6

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

$\frac{\frac{(m + n) (m - n)}{m n} \cdot 4 m n}{(m + n) (m - n)}$

Step 7.6

Combine exponents.

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Step 7.6.1

Combine $\frac{(m + n) (m - n)}{m n}$ and $4$ .

$\frac{\frac{(m + n) (m - n) \cdot 4}{m n} m n}{(m + n) (m - n)}$

Step 7.6.2

Combine $\frac{(m + n) (m - n) \cdot 4}{m n}$ and $m$ .

$\frac{\frac{(m + n) (m - n) \cdot 4 m}{m n} n}{(m + n) (m - n)}$

Step 7.6.3

Combine $\frac{(m + n) (m - n) \cdot 4 m}{m n}$ and $n$ .

$\frac{\frac{(m + n) (m - n) \cdot 4 m n}{m n}}{(m + n) (m - n)}$

Step 7.7

Reduce the expression $\frac{(m + n) (m - n) \cdot 4 m n}{m n}$ by cancelling the common factors.

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Step 7.7.1

Cancel the common factor.

$\frac{\frac{(m + n) (m - n) \cdot 4 m n}{m n}}{(m + n) (m - n)}$

Step 7.7.2

Rewrite the expression.

$\frac{\frac{(m + n) (m - n) \cdot 4 n}{n}}{(m + n) (m - n)}$

Step 7.8

Cancel the common factor of $n$ .

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Step 7.8.1

Cancel the common factor.

$\frac{\frac{(m + n) (m - n) \cdot 4 n}{n}}{(m + n) (m - n)}$

Step 7.8.2

Divide $(m + n) (m - n) \cdot 4$ by $1$ .

$\frac{(m + n) (m - n) \cdot 4}{(m + n) (m - n)}$

Step 7.9

Expand $(m + n) (m - n)$ using the FOIL Method.

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Step 7.9.1

Apply the distributive property.

$\frac{(m (m - n) + n (m - n)) \cdot 4}{(m + n) (m - n)}$

Step 7.9.2

Apply the distributive property.

$\frac{(m \cdot m + m (- n) + n (m - n)) \cdot 4}{(m + n) (m - n)}$

Step 7.9.3

Apply the distributive property.

$\frac{(m \cdot m + m (- n) + n m + n (- n)) \cdot 4}{(m + n) (m - n)}$

Step 7.10

Combine the opposite terms in $m \cdot m + m (- n) + n m + n (- n)$ .

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Step 7.10.1

Reorder the factors in the terms $m (- n)$ and $n m$ .

$\frac{(m \cdot m - m n + m n + n (- n)) \cdot 4}{(m + n) (m - n)}$

Step 7.10.2

Add $- m n$ and $m n$ .

$\frac{(m \cdot m + 0 + n (- n)) \cdot 4}{(m + n) (m - n)}$

Step 7.10.3

Add $m \cdot m$ and $0$ .

$\frac{(m \cdot m + n (- n)) \cdot 4}{(m + n) (m - n)}$

Step 7.11

Simplify each term.

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Step 7.11.1

Multiply $m$ by $m$ .

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

Step 7.11.2

Rewrite using the commutative property of multiplication.

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

Step 7.11.3

Multiply $n$ by $n$ by adding the exponents.

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Step 7.11.3.1

Move $n$ .

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

Step 7.11.3.2

Multiply $n$ by $n$ .

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

Step 7.12

Apply the distributive property.

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

Step 7.13

Move $4$ to the left of $m^{2}$ .

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

Step 7.14

Multiply $4$ by $- 1$ .

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

Step 7.15

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

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Step 7.15.1

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

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

Step 7.15.2

Factor $4$ out of $- 4 n^{2}$ .

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

Step 7.15.3

Factor $4$ out of $4 (m^{2}) + 4 (- n^{2})$ .

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

Step 7.16

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

$\frac{4 (m + n) (m - n)}{(m + n) (m - n)}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $m + n$ .

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Step 8.1.1

Cancel the common factor.

$\frac{4 (m + n) (m - n)}{(m + n) (m - n)}$

Step 8.1.2

Rewrite the expression.

$\frac{4 (m - n)}{m - n}$

Step 8.2

Cancel the common factor of $m - n$ .

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Step 8.2.1

Cancel the common factor.

$\frac{4 (m - n)}{m - n}$

Step 8.2.2

Divide $4$ by $1$ .

$4$