Basic Math Examples

$\frac{n^{2} - m^{2}}{2 m - 3 n} \div \frac{m - n}{4 m^{2} - 9 n^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{n^{2} - m^{2}}{2 m - 3 n} \cdot \frac{4 m^{2} - 9 n^{2}}{m - n}$

Step 2

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

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{4 m^{2} - 9 n^{2}}{m - n}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $4 m^{2}$ as ${(2 m)}^{2}$ .

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{{(2 m)}^{2} - 9 n^{2}}{m - n}$

Step 3.2

Rewrite $9 n^{2}$ as ${(3 n)}^{2}$ .

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{{(2 m)}^{2} - {(3 n)}^{2}}{m - n}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 m$ and $b = 3 n$ .

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{(2 m + 3 n) (2 m - (3 n))}{m - n}$

Step 3.4

Multiply $3$ by $- 1$ .

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{(2 m + 3 n) (2 m - 3 n)}{m - n}$

Step 4

Cancel the common factor of $2 m - 3 n$ .

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Step 4.1

Factor $2 m - 3 n$ out of $(2 m + 3 n) (2 m - 3 n)$ .

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{(2 m - 3 n) (2 m + 3 n)}{m - n}$

Step 4.2

Cancel the common factor.

$\frac{(n + m) (n - m)}{2 m - 3 n} \cdot \frac{(2 m - 3 n) (2 m + 3 n)}{m - n}$

Step 4.3

Rewrite the expression.

$(n + m) (n - m) \frac{2 m + 3 n}{m - n}$

Step 5

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

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Step 5.1

Apply the distributive property.

$(n (n - m) + m (n - m)) \frac{2 m + 3 n}{m - n}$

Step 5.2

Apply the distributive property.

$(n \cdot n + n (- m) + m (n - m)) \frac{2 m + 3 n}{m - n}$

Step 5.3

Apply the distributive property.

$(n \cdot n + n (- m) + m n + m (- m)) \frac{2 m + 3 n}{m - n}$

Step 6

Simplify terms.

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Step 6.1

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

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Step 6.1.1

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

$(n \cdot n - m n + m n + m (- m)) \frac{2 m + 3 n}{m - n}$

Step 6.1.2

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

$(n \cdot n + 0 + m (- m)) \frac{2 m + 3 n}{m - n}$

Step 6.1.3

Add $n \cdot n$ and $0$ .

$(n \cdot n + m (- m)) \frac{2 m + 3 n}{m - n}$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $n$ by $n$ .

$(n^{2} + m (- m)) \frac{2 m + 3 n}{m - n}$

Step 6.2.2

Rewrite using the commutative property of multiplication.

$(n^{2} - m \cdot m) \frac{2 m + 3 n}{m - n}$

Step 6.2.3

Multiply $m$ by $m$ by adding the exponents.

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Step 6.2.3.1

Move $m$ .

$(n^{2} - (m \cdot m)) \frac{2 m + 3 n}{m - n}$

Step 6.2.3.2

Multiply $m$ by $m$ .

$(n^{2} - m^{2}) \frac{2 m + 3 n}{m - n}$

Step 6.3

Multiply $n^{2} - m^{2}$ by $\frac{2 m + 3 n}{m - n}$ .

$\frac{(n^{2} - m^{2}) (2 m + 3 n)}{m - n}$

Step 7

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

$\frac{(n + m) (n - m) (2 m + 3 n)}{m - n}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $n - m$ and $m - n$ .

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Step 8.1.1

Factor $- 1$ out of $n$ .

$\frac{(n + m) (- 1 (- n) - m) (2 m + 3 n)}{m - n}$

Step 8.1.2

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

$\frac{(n + m) (- 1 (- n) - (m)) (2 m + 3 n)}{m - n}$

Step 8.1.3

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

$\frac{(n + m) (- 1 (- n + m)) (2 m + 3 n)}{m - n}$

Step 8.1.4

Reorder terms.

$\frac{(n + m) (- 1 (m - n)) (2 m + 3 n)}{m - n}$

Step 8.1.5

Cancel the common factor.

$\frac{(n + m) (- 1 (m - n)) (2 m + 3 n)}{m - n}$

Step 8.1.6

Divide $((n + m) \cdot (- 1)) (2 m + 3 n)$ by $1$ .

$((n + m) \cdot (- 1)) (2 m + 3 n)$

Step 8.2

Simplify by multiplying through.

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Step 8.2.1

Apply the distributive property.

$(n \cdot - 1 + m \cdot - 1) (2 m + 3 n)$

Step 8.2.2

Reorder.

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Step 8.2.2.1

Move $- 1$ to the left of $n$ .

$(- 1 \cdot n + m \cdot - 1) (2 m + 3 n)$

Step 8.2.2.2

Move $- 1$ to the left of $m$ .

$(- 1 \cdot n - 1 \cdot m) (2 m + 3 n)$

Step 8.3

Simplify each term.

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Step 8.3.1

Rewrite $- 1 n$ as $- n$ .

$(- n - 1 \cdot m) (2 m + 3 n)$

Step 8.3.2

Rewrite $- 1 m$ as $- m$ .

$(- n - m) (2 m + 3 n)$

Step 9

Expand $(- n - m) (2 m + 3 n)$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

$- n (2 m + 3 n) - m (2 m + 3 n)$

Step 9.2

Apply the distributive property.

$- n (2 m) - n (3 n) - m (2 m + 3 n)$

Step 9.3

Apply the distributive property.

$- n (2 m) - n (3 n) - m (2 m) - m (3 n)$

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot 2 n m - n (3 n) - m (2 m) - m (3 n)$

Step 10.1.2

Multiply $- 1$ by $2$ .

$- 2 n m - n (3 n) - m (2 m) - m (3 n)$

Step 10.1.3

Rewrite using the commutative property of multiplication.

$- 2 n m - 1 \cdot 3 n \cdot n - m (2 m) - m (3 n)$

Step 10.1.4

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

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Step 10.1.4.1

Move $n$ .

$- 2 n m - 1 \cdot 3 (n \cdot n) - m (2 m) - m (3 n)$

Step 10.1.4.2

Multiply $n$ by $n$ .

$- 2 n m - 1 \cdot 3 n^{2} - m (2 m) - m (3 n)$

Step 10.1.5

Multiply $- 1$ by $3$ .

$- 2 n m - 3 n^{2} - m (2 m) - m (3 n)$

Step 10.1.6

Rewrite using the commutative property of multiplication.

$- 2 n m - 3 n^{2} - 1 \cdot 2 m \cdot m - m (3 n)$

Step 10.1.7

Multiply $m$ by $m$ by adding the exponents.

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Step 10.1.7.1

Move $m$ .

$- 2 n m - 3 n^{2} - 1 \cdot 2 (m \cdot m) - m (3 n)$

Step 10.1.7.2

Multiply $m$ by $m$ .

$- 2 n m - 3 n^{2} - 1 \cdot 2 m^{2} - m (3 n)$

Step 10.1.8

Multiply $- 1$ by $2$ .

$- 2 n m - 3 n^{2} - 2 m^{2} - m (3 n)$

Step 10.1.9

Rewrite using the commutative property of multiplication.

$- 2 n m - 3 n^{2} - 2 m^{2} - 1 \cdot 3 m n$

Step 10.1.10

Multiply $- 1$ by $3$ .

$- 2 n m - 3 n^{2} - 2 m^{2} - 3 m n$

Step 10.2

Subtract $3 m n$ from $- 2 n m$ .

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Step 10.2.1

Move $n$ .

$- 3 n^{2} - 2 m^{2} - 2 m n - 3 m n$

Step 10.2.2

Subtract $3 m n$ from $- 2 m n$ .

$- 3 n^{2} - 2 m^{2} - 5 m n$

Step 11

Move $- 3 n^{2}$ .

$- 2 m^{2} - 5 m n - 3 n^{2}$