Basic Math Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 m n = 2 \cdot m \cdot n$

Step 1.2

Rewrite the polynomial.

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

Step 1.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = m$ and $b = n$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Factor $2 m$ out of $2 m n$ .

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

Step 2.3

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

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

Step 3

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 $b = n$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $m + n$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Multiply $\frac{m + n}{2 m (m + n)}$ by $\frac{m - n}{m - n}$ .

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

Step 4.3

Cancel the common factor of $m + n$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $m - n$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

$\frac{1}{2 m}$