Basic Math Examples

$\frac{\frac{z^{2} - w^{2}}{z^{3} - w^{3}}}{\frac{z^{2} + z w + w^{2}}{z^{2} + 2 z w + w^{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{z^{2} - w^{2}}{z^{3} - w^{3}} \cdot \frac{z^{2} + 2 z w + w^{2}}{z^{2} + z w + w^{2}}$

Step 2

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

$\frac{(z + w) (z - w)}{z^{3} - w^{3}} \cdot \frac{z^{2} + 2 z w + w^{2}}{z^{2} + z w + w^{2}}$

Step 3

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 $a = z$ and $b = w$ .

$\frac{(z + w) (z - w)}{(z - w) (z^{2} + z w + w^{2})} \cdot \frac{z^{2} + 2 z w + w^{2}}{z^{2} + z w + w^{2}}$

Step 4

Cancel the common factor of $z - w$ .

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Step 4.1

Cancel the common factor.

$\frac{(z + w) (z - w)}{(z - w) (z^{2} + z w + w^{2})} \cdot \frac{z^{2} + 2 z w + w^{2}}{z^{2} + z w + w^{2}}$

Step 4.2

Rewrite the expression.

$\frac{z + w}{z^{2} + z w + w^{2}} \cdot \frac{z^{2} + 2 z w + w^{2}}{z^{2} + z w + w^{2}}$

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.

$2 z w = 2 \cdot z \cdot w$

Step 5.2

Rewrite the polynomial.

$\frac{z + w}{z^{2} + z w + w^{2}} \cdot \frac{z^{2} + 2 \cdot z \cdot w + w^{2}}{z^{2} + z w + w^{2}}$

Step 5.3

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

$\frac{z + w}{z^{2} + z w + w^{2}} \cdot \frac{{(z + w)}^{2}}{z^{2} + z w + w^{2}}$

Step 6

Multiply $\frac{z + w}{z^{2} + z w + w^{2}} \cdot \frac{{(z + w)}^{2}}{z^{2} + z w + w^{2}}$ .

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Step 6.1

Multiply $\frac{z + w}{z^{2} + z w + w^{2}}$ by $\frac{{(z + w)}^{2}}{z^{2} + z w + w^{2}}$ .

$\frac{(z + w) {(z + w)}^{2}}{(z^{2} + z w + w^{2}) (z^{2} + z w + w^{2})}$

Step 6.2

Multiply $z + w$ by ${(z + w)}^{2}$ by adding the exponents.

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Step 6.2.1

Multiply $z + w$ by ${(z + w)}^{2}$ .

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Step 6.2.1.1

Raise $z + w$ to the power of $1$ .

$\frac{{(z + w)}^{1} {(z + w)}^{2}}{(z^{2} + z w + w^{2}) (z^{2} + z w + w^{2})}$

Step 6.2.1.2

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

$\frac{{(z + w)}^{1 + 2}}{(z^{2} + z w + w^{2}) (z^{2} + z w + w^{2})}$

Step 6.2.2

Add $1$ and $2$ .

$\frac{{(z + w)}^{3}}{(z^{2} + z w + w^{2}) (z^{2} + z w + w^{2})}$

Step 6.3

Raise $z^{2} + z w + w^{2}$ to the power of $1$ .

$\frac{{(z + w)}^{3}}{{(z^{2} + z w + w^{2})}^{1} (z^{2} + z w + w^{2})}$

Step 6.4

Raise $z^{2} + z w + w^{2}$ to the power of $1$ .

$\frac{{(z + w)}^{3}}{{(z^{2} + z w + w^{2})}^{1} {(z^{2} + z w + w^{2})}^{1}}$

Step 6.5

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

$\frac{{(z + w)}^{3}}{{(z^{2} + z w + w^{2})}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$\frac{{(z + w)}^{3}}{{(z^{2} + z w + w^{2})}^{2}}$