Basic Math Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $1$ as $1^{3}$ .

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

Step 2.2

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 $a = w$ and $b = 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Multiply $w$ by $1$ .

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

Step 2.3.2

One to any power is one.

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

Step 3

Cancel the common factors.

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

$\frac{w^{2} + w + 1}{w - 1} \cdot \frac{w^{2} + w + 1}{w^{2} - 1}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{w^{2} + w + 1}{w - 1} \cdot \frac{w^{2} + w + 1}{w^{2} - 1^{2}}$

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 $a = w$ and $b = 1$ .

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

Step 5

Multiply $\frac{w^{2} + w + 1}{w - 1} \cdot \frac{w^{2} + w + 1}{(w + 1) (w - 1)}$ .

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Step 5.1

Multiply $\frac{w^{2} + w + 1}{w - 1}$ by $\frac{w^{2} + w + 1}{(w + 1) (w - 1)}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

Raise $w - 1$ to the power of $1$ .

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

Step 5.7

Raise $w - 1$ to the power of $1$ .

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

Step 5.8

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

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

Step 5.9

Add $1$ and $1$ .

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