Basic Math Examples

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

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

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 n \cdot m

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

Step 1.2

Rewrite the polynomial.

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

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

Step 1.3

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = n

$a = n$ and

b = m

$b = m$ .

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

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

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

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

Step 2

Rewrite

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

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

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Step 2.1

Rewrite

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

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

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

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

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

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = n

$a = n$ and

b = \frac{m}{n - m}

$b = \frac{m}{n - m}$ .

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

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

Step 2.3

Simplify.

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Step 2.3.1

To write

n

$n$ as a fraction with a common denominator, multiply by

\frac{n - m}{n - m}

$\frac{n - m}{n - m}$ .

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

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

Step 2.3.2

Combine the numerators over the common denominator.

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

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

Step 2.3.3

Rewrite

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

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

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Step 2.3.3.1

Apply the distributive property.

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

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

Step 2.3.3.2

Multiply

n

$n$ by

n

$n$ .

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

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

Step 2.3.3.3

Rewrite using the commutative property of multiplication.

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

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

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

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

Step 2.3.4

To write

n

$n$ as a fraction with a common denominator, multiply by

\frac{n - m}{n - m}

$\frac{n - m}{n - m}$ .

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

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

Step 2.3.5

Combine the numerators over the common denominator.

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

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

Step 2.3.6

Rewrite

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

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

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Step 2.3.6.1

Apply the distributive property.

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

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

Step 2.3.6.2

Multiply

n

$n$ by

n

$n$ .

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

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

Step 2.3.6.3

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

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

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