Basic Math Examples

$\frac{y^{3} - 27}{y^{2} + y - 56} \cdot \frac{y^{2} + 11 y + 24}{y^{2} - 9}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $27$ as $3^{3}$ .

$\frac{y^{3} - 3^{3}}{y^{2} + y - 56} \cdot \frac{y^{2} + 11 y + 24}{y^{2} - 9}$

Step 1.2

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

$\frac{(y - 3) (y^{2} + y \cdot 3 + 3^{2})}{y^{2} + y - 56} \cdot \frac{y^{2} + 11 y + 24}{y^{2} - 9}$

Step 1.3

Simplify.

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Step 1.3.1

Move $3$ to the left of $y$ .

$\frac{(y - 3) (y^{2} + 3 \cdot y + 3^{2})}{y^{2} + y - 56} \cdot \frac{y^{2} + 11 y + 24}{y^{2} - 9}$

Step 1.3.2

Raise $3$ to the power of $2$ .

$\frac{(y - 3) (y^{2} + 3 y + 9)}{y^{2} + y - 56} \cdot \frac{y^{2} + 11 y + 24}{y^{2} - 9}$

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.

$\frac{(y - 3) (y^{2} + 3 y + 9)}{(y - 7) (y + 8)} \cdot \frac{y^{2} + 11 y + 24}{y^{2} - 9}$

Step 3

Factor $y^{2} + 11 y + 24$ 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 $24$ and whose sum is $11$ .

$3, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(y - 3) (y^{2} + 3 y + 9)}{(y - 7) (y + 8)} \cdot \frac{(y + 3) (y + 8)}{y^{2} - 9}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$\frac{(y - 3) (y^{2} + 3 y + 9)}{(y - 7) (y + 8)} \cdot \frac{(y + 3) (y + 8)}{y^{2} - 3^{2}}$

Step 4.2

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

$\frac{(y - 3) (y^{2} + 3 y + 9)}{(y - 7) (y + 8)} \cdot \frac{(y + 3) (y + 8)}{(y + 3) (y - 3)}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{(y - 3) (y^{2} + 3 y + 9) ((y + 3) (y + 8))}{(y - 7) (y + 8) ((y + 3) (y - 3))}$

Step 5.2

Cancel the common factor of $y - 3$ .

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Step 5.2.1

Cancel the common factor.

$\frac{(y - 3) (y^{2} + 3 y + 9) ((y + 3) (y + 8))}{(y - 7) (y + 8) ((y + 3) (y - 3))}$

Step 5.2.2

Rewrite the expression.

$\frac{(y^{2} + 3 y + 9) ((y + 3) (y + 8))}{(y - 7) (y + 8) (y + 3)}$

Step 5.3

Cancel the common factor of $y + 3$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(y^{2} + 3 y + 9) ((y + 3) (y + 8))}{(y - 7) (y + 8) (y + 3)}$

Step 5.3.2

Rewrite the expression.

$\frac{(y^{2} + 3 y + 9) (y + 8)}{(y - 7) (y + 8)}$

Step 5.4

Cancel the common factor of $y + 8$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(y^{2} + 3 y + 9) (y + 8)}{(y - 7) (y + 8)}$

Step 5.4.2

Rewrite the expression.

$\frac{y^{2} + 3 y + 9}{y - 7}$