Basic Math Examples

$(\frac{15 y}{y^{2} - 4 y + 4}) (\frac{y^{2} - 4}{30 y^{2}})$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{15 y}{y^{2} - 4 y + 2^{2}} \cdot \frac{y^{2} - 4}{30 y^{2}}$

Step 1.2

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

$4 y = 2 \cdot y \cdot 2$

Step 1.3

Rewrite the polynomial.

$\frac{15 y}{y^{2} - 2 \cdot y \cdot 2 + 2^{2}} \cdot \frac{y^{2} - 4}{30 y^{2}}$

Step 1.4

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

$\frac{15 y}{{(y - 2)}^{2}} \cdot \frac{y^{2} - 4}{30 y^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $4$ as $2^{2}$ .

$\frac{15 y}{{(y - 2)}^{2}} \cdot \frac{y^{2} - 2^{2}}{30 y^{2}}$

Step 2.2

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 $b = 2$ .

$\frac{15 y}{{(y - 2)}^{2}} \cdot \frac{(y + 2) (y - 2)}{30 y^{2}}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{15 y ((y + 2) (y - 2))}{{(y - 2)}^{2} (30 y^{2})}$

Step 3.2

Cancel the common factor of $15$ and $30$ .

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Step 3.2.1

Factor $15$ out of $15 y ((y + 2) (y - 2))$ .

$\frac{15 (y ((y + 2) (y - 2)))}{{(y - 2)}^{2} (30 y^{2})}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $15$ out of ${(y - 2)}^{2} (30 y^{2})$ .

$\frac{15 (y ((y + 2) (y - 2)))}{15 ({(y - 2)}^{2} (2 y^{2}))}$

Step 3.2.2.2

Cancel the common factor.

$\frac{15 (y ((y + 2) (y - 2)))}{15 ({(y - 2)}^{2} (2 y^{2}))}$

Step 3.2.2.3

Rewrite the expression.

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

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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Step 5.1

Factor $y - 2$ out of $(y + 2) (y - 2)$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

$\frac{(y - 2) (y + 2)}{(y - 2) ((y - 2) (2 y))}$

Step 5.2.2

Cancel the common factor.

$\frac{(y - 2) (y + 2)}{(y - 2) ((y - 2) (2 y))}$

Step 5.2.3

Rewrite the expression.

$\frac{y + 2}{(y - 2) (2 y)}$

Step 6

Simplify the expression.

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Step 6.1

Move $2$ to the left of $y - 2$ .

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

Step 6.2

Reorder factors in $\frac{y + 2}{2 (y - 2) y}$ .

$\frac{y + 2}{2 y (y - 2)}$