Algebra Examples

$(1 - r^{3}) {(1 - r)}^{- 3}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(1 - r^{3}) \frac{1}{{(1 - r)}^{3}}$

Step 2

Multiply $1 - r^{3}$ by $\frac{1}{{(1 - r)}^{3}}$ .

$\frac{1 - r^{3}}{{(1 - r)}^{3}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{1^{3} - r^{3}}{{(1 - r)}^{3}}$

Step 3.2

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 $a = 1$ and $b = r$ .

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

Step 3.3

Simplify.

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Step 3.3.1

One to any power is one.

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

Step 3.3.2

Multiply $r$ by $1$ .

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $1 - r$ out of ${(1 - r)}^{3}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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