Algebra Examples

$\frac{36 y}{y^{2} - 2 y + 1} \div \frac{12 y^{2}}{y^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{36 y}{y^{2} - 2 y + 1} \cdot \frac{y^{2} - 1}{12 y^{2}}$

Step 2

Cancel the common factor of $12 y$ .

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Step 2.1

Factor $12 y$ out of $36 y$ .

$\frac{12 y (3)}{y^{2} - 2 y + 1} \cdot \frac{y^{2} - 1}{12 y^{2}}$

Step 2.2

Factor $12 y$ out of $12 y^{2}$ .

$\frac{12 y (3)}{y^{2} - 2 y + 1} \cdot \frac{y^{2} - 1}{12 y (y)}$

Step 2.3

Cancel the common factor.

$\frac{12 y \cdot 3}{y^{2} - 2 y + 1} \cdot \frac{y^{2} - 1}{12 y \cdot y}$

Step 2.4

Rewrite the expression.

$\frac{3}{y^{2} - 2 y + 1} \cdot \frac{y^{2} - 1}{y}$

Step 3

Multiply $\frac{3}{y^{2} - 2 y + 1}$ by $\frac{y^{2} - 1}{y}$ .

$\frac{3 (y^{2} - 1)}{(y^{2} - 2 y + 1) y}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 (y^{2} - 1^{2})}{(y^{2} - 2 y + 1) y}$

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 $b = 1$ .

$\frac{3 (y + 1) (y - 1)}{(y^{2} - 2 y + 1) y}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 (y + 1) (y - 1)}{(y^{2} - 2 y + 1^{2}) y}$

Step 5.2

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

$2 y = 2 \cdot y \cdot 1$

Step 5.3

Rewrite the polynomial.

$\frac{3 (y + 1) (y - 1)}{(y^{2} - 2 \cdot y \cdot 1 + 1^{2}) y}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 1$ .

$\frac{3 (y + 1) (y - 1)}{{(y - 1)}^{2} y}$

Step 6

Cancel the common factor of $y - 1$ and ${(y - 1)}^{2}$ .

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Step 6.1

Factor $y - 1$ out of $3 (y + 1) (y - 1)$ .

$\frac{(y - 1) (3 (y + 1))}{{(y - 1)}^{2} y}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $y - 1$ out of ${(y - 1)}^{2} y$ .

$\frac{(y - 1) (3 (y + 1))}{(y - 1) ((y - 1) y)}$

Step 6.2.2

Cancel the common factor.

$\frac{(y - 1) (3 (y + 1))}{(y - 1) ((y - 1) y)}$

Step 6.2.3

Rewrite the expression.

$\frac{3 (y + 1)}{(y - 1) y}$

Step 7

Reorder factors in $\frac{3 (y + 1)}{(y - 1) y}$ .

$\frac{3 (y + 1)}{y (y - 1)}$

Step 8

Apply the distributive property.

$\frac{3 y + 3 \cdot 1}{y (y - 1)}$

Step 9

Multiply $3$ by $1$ .

$\frac{3 y + 3}{y (y - 1)}$

Step 10

Split the fraction $\frac{3 y + 3}{y (y - 1)}$ into two fractions.

$\frac{3 y}{y (y - 1)} + \frac{3}{y (y - 1)}$

Step 11

Cancel the common factor of $y$ .

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Step 11.1

Cancel the common factor.

$\frac{3 y}{y (y - 1)} + \frac{3}{y (y - 1)}$

Step 11.2

Rewrite the expression.

$\frac{3}{y - 1} + \frac{3}{y (y - 1)}$