Basic Math Examples

\frac{b^{2 - 25}}{b^{2} + b - 10} \div \frac{5 - b}{b^{2} + 5 b - 14}

$\frac{b^{2 - 25}}{b^{2} + b - 10} \div \frac{5 - b}{b^{2} + 5 b - 14}$

Step 1

To divide by a fraction, multiply by its reciprocal.

\frac{b^{2 - 25}}{b^{2} + b - 10} \cdot \frac{b^{2} + 5 b - 14}{5 - b}

$\frac{b^{2 - 25}}{b^{2} + b - 10} \cdot \frac{b^{2} + 5 b - 14}{5 - b}$

Step 2

Move

b^{2 - 25}

$b^{2 - 25}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{1}{(b^{2} + b - 10) b^{- (2 - 25)}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}

$\frac{1}{(b^{2} + b - 10) b^{- (2 - 25)}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}$

Step 3

Simplify the denominator.

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Step 3.1

Subtract

25

$25$ from

2

$2$ .

\frac{1}{(b^{2} + b - 10) b^{- - 23}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}

$\frac{1}{(b^{2} + b - 10) b^{- - 23}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}$

Step 3.2

Multiply

- 1

$- 1$ by

- 23

$- 23$ .

\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}

$\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}$

\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}

$\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{b^{2} + 5 b - 14}{5 - b}$

Step 4

Factor

b^{2} + 5 b - 14

$b^{2} + 5 b - 14$ using the AC method.

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Step 4.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 14

$- 14$ and whose sum is

5

$5$ .

- 2, 7

$- 2, 7$

Step 4.2

Write the factored form using these integers.

\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{(b - 2) (b + 7)}{5 - b}

$\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{(b - 2) (b + 7)}{5 - b}$

\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{(b - 2) (b + 7)}{5 - b}

$\frac{1}{(b^{2} + b - 10) b^{23}} \cdot \frac{(b - 2) (b + 7)}{5 - b}$

Step 5

Combine fractions.

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Step 5.1

Multiply

\frac{1}{(b^{2} + b - 10) b^{23}}

$\frac{1}{(b^{2} + b - 10) b^{23}}$ by

\frac{(b - 2) (b + 7)}{5 - b}

$\frac{(b - 2) (b + 7)}{5 - b}$ .

$\frac{(b - 2) (b + 7)}{(b^{2} + b - 10) b^{23} (5 - b)}$

Step 5.2

Reorder factors in

$\frac{(b - 2) (b + 7)}{(b^{2} + b - 10) b^{23} (5 - b)}$ .

$\frac{(b - 2) (b + 7)}{b^{23} (b^{2} + b - 10) (5 - b)}$