Basic Math Examples

$\frac{5}{y + 7} \div \frac{15 y}{y^{2} + 14 y + 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{5}{y + 7} \cdot \frac{y^{2} + 14 y + 49}{15 y}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $5$ .

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Step 2.1.1

Factor $5$ out of $15 y$ .

$\frac{5}{y + 7} \cdot \frac{y^{2} + 14 y + 49}{5 (3 y)}$

Step 2.1.2

Cancel the common factor.

$\frac{5}{y + 7} \cdot \frac{y^{2} + 14 y + 49}{5 (3 y)}$

Step 2.1.3

Rewrite the expression.

$\frac{1}{y + 7} \cdot \frac{y^{2} + 14 y + 49}{3 y}$

Step 2.2

Multiply $\frac{1}{y + 7}$ by $\frac{y^{2} + 14 y + 49}{3 y}$ .

$\frac{y^{2} + 14 y + 49}{(y + 7) (3 y)}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $49$ as $7^{2}$ .

$\frac{y^{2} + 14 y + 7^{2}}{(y + 7) \cdot 3 y}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 y = 2 \cdot y \cdot 7$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 7$ .

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

Step 4

Cancel the common factor of ${(y + 7)}^{2}$ and $y + 7$ .

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Step 4.1

Factor $y + 7$ out of ${(y + 7)}^{2}$ .

$\frac{(y + 7) (y + 7)}{(y + 7) \cdot 3 y}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $y + 7$ out of $(y + 7) \cdot 3 y$ .

$\frac{(y + 7) (y + 7)}{(y + 7) (3 y)}$

Step 4.2.2

Cancel the common factor.

$\frac{(y + 7) (y + 7)}{(y + 7) (3 y)}$

Step 4.2.3

Rewrite the expression.

$\frac{y + 7}{3 y}$