Basic Math Examples

$\frac{y^{2} + y - 56}{y^{2} + y - 12} \cdot \frac{y^{2} + 13 y + 40}{y^{2} - y - 42} \div \frac{y + 8}{y^{2} + 10 y + 24}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} + y - 56}{y^{2} + y - 12} \cdot \frac{y^{2} + 13 y + 40}{y^{2} - y - 42} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 2

Factor $y^{2} + y - 56$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 56$ and whose sum is $1$ .

$- 7, 8$

Step 2.2

Write the factored form using these integers.

$\frac{(y - 7) (y + 8)}{y^{2} + y - 12} \cdot \frac{y^{2} + 13 y + 40}{y^{2} - y - 42} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 3

Factor $y^{2} + y - 12$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(y - 7) (y + 8)}{(y - 3) (y + 4)} \cdot \frac{y^{2} + 13 y + 40}{y^{2} - y - 42} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 4

Factor $y^{2} + 13 y + 40$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $40$ and whose sum is $13$ .

$5, 8$

Step 4.2

Write the factored form using these integers.

$\frac{(y - 7) (y + 8)}{(y - 3) (y + 4)} \cdot \frac{(y + 5) (y + 8)}{y^{2} - y - 42} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 5

Factor $y^{2} - y - 42$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 5.2

Write the factored form using these integers.

$\frac{(y - 7) (y + 8)}{(y - 3) (y + 4)} \cdot \frac{(y + 5) (y + 8)}{(y - 7) (y + 6)} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $y - 7$ .

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Step 6.1.1

Cancel the common factor.

$\frac{(y - 7) (y + 8)}{(y - 3) (y + 4)} \cdot \frac{(y + 5) (y + 8)}{(y - 7) (y + 6)} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 6.1.2

Rewrite the expression.

$\frac{y + 8}{(y - 3) (y + 4)} \cdot \frac{(y + 5) (y + 8)}{y + 6} \frac{y^{2} + 10 y + 24}{y + 8}$

Step 6.2

Multiply $\frac{y + 8}{(y - 3) (y + 4)}$ by $\frac{(y + 5) (y + 8)}{y + 6}$ .

$\frac{(y + 8) ((y + 5) (y + 8))}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 7

Raise $y + 8$ to the power of $1$ .

$\frac{{(y + 8)}^{1} (y + 8) (y + 5)}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 8

Raise $y + 8$ to the power of $1$ .

$\frac{{(y + 8)}^{1} {(y + 8)}^{1} (y + 5)}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 9

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

$\frac{{(y + 8)}^{1 + 1} (y + 5)}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 10

Add $1$ and $1$ .

$\frac{{(y + 8)}^{2} (y + 5)}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 11

Cancel the common factor of $y + 8$ .

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Step 11.1

Factor $y + 8$ out of ${(y + 8)}^{2} (y + 5)$ .

$\frac{(y + 8) ((y + 8) (y + 5))}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 11.2

Cancel the common factor.

$\frac{(y + 8) ((y + 8) (y + 5))}{(y - 3) (y + 4) (y + 6)} \cdot \frac{y^{2} + 10 y + 24}{y + 8}$

Step 11.3

Rewrite the expression.

$\frac{(y + 8) (y + 5)}{(y - 3) (y + 4) (y + 6)} (y^{2} + 10 y + 24)$

Step 12

Multiply $\frac{(y + 8) (y + 5)}{(y - 3) (y + 4) (y + 6)}$ by $y^{2} + 10 y + 24$ .

$\frac{(y + 8) (y + 5) (y^{2} + 10 y + 24)}{(y - 3) (y + 4) (y + 6)}$

Step 13

Factor $y^{2} + 10 y + 24$ using the AC method.

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Step 13.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $24$ and whose sum is $10$ .

$4, 6$

Step 13.2

Write the factored form using these integers.

$\frac{(y + 8) (y + 5) ((y + 4) (y + 6))}{(y - 3) (y + 4) (y + 6)}$

$\frac{(y + 8) (y + 5) (y + 4) (y + 6)}{(y - 3) (y + 4) (y + 6)}$

Step 14

Cancel the common factor of $y + 4$ .

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Step 14.1

Cancel the common factor.

$\frac{(y + 8) (y + 5) (y + 4) (y + 6)}{(y - 3) (y + 4) (y + 6)}$

Step 14.2

Rewrite the expression.

$\frac{(y + 8) (y + 5) (y + 6)}{(y - 3) (y + 6)}$

Step 15

Cancel the common factor of $y + 6$ .

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Step 15.1

Cancel the common factor.

$\frac{(y + 8) (y + 5) (y + 6)}{(y - 3) (y + 6)}$

Step 15.2

Rewrite the expression.

$\frac{(y + 8) (y + 5)}{y - 3}$