Basic Math Examples

$\frac{\frac{a^{3} - y^{3}}{a^{2} - 2 a y + y^{2}}}{\frac{a^{2} - y^{2}}{a^{2} + 2 a y + y^{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 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 $a = a$ and $b = y$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$2 a y = 2 \cdot a \cdot y$

Step 3.2

Rewrite the polynomial.

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

Step 3.3

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

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

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\frac{a^{2} + a y + y^{2}}{a - y} \cdot \frac{a^{2} + 2 a y + y^{2}}{a^{2} - y^{2}}$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

$2 a y = 2 \cdot a \cdot y$

Step 5.2

Rewrite the polynomial.

$\frac{a^{2} + a y + y^{2}}{a - y} \cdot \frac{a^{2} + 2 \cdot a \cdot y + y^{2}}{a^{2} - y^{2}}$

Step 5.3

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$\frac{a^{2} + a y + y^{2}}{a - y} \cdot \frac{a + y}{a - y}$

Step 8

Multiply $\frac{a^{2} + a y + y^{2}}{a - y} \cdot \frac{a + y}{a - y}$ .

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Step 8.1

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

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

Step 8.2

Raise $a - y$ to the power of $1$ .

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

Step 8.3

Raise $a - y$ to the power of $1$ .

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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