Basic Math Examples

$\frac{p^{k} - 1}{p - 1} + p^{k}$

Step 1

To write $p^{k}$ as a fraction with a common denominator, multiply by $\frac{p - 1}{p - 1}$ .

$\frac{p^{k} - 1}{p - 1} + \frac{p^{k} (p - 1)}{p - 1}$

Step 2

Combine the numerators over the common denominator.

$\frac{p^{k} - 1 + p^{k} (p - 1)}{p - 1}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

$\frac{p^{k} - 1 + p^{k} p + p^{k} \cdot - 1}{p - 1}$

Step 3.2

Multiply $p^{k}$ by $p$ .

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Step 3.2.1

Raise $p$ to the power of $1$ .

$\frac{p^{k} - 1 + p^{k} p^{1} + p^{k} \cdot - 1}{p - 1}$

Step 3.2.2

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

$\frac{p^{k} - 1 + p^{k + 1} + p^{k} \cdot - 1}{p - 1}$

Step 3.3

Move $- 1$ to the left of $p^{k}$ .

$\frac{p^{k} - 1 + p^{k + 1} - 1 \cdot p^{k}}{p - 1}$

Step 3.4

Rewrite $- 1 p^{k}$ as $- p^{k}$ .

$\frac{p^{k} - 1 + p^{k + 1} - p^{k}}{p - 1}$

Step 3.5

Subtract $p^{k}$ from $p^{k}$ .

$\frac{- 1 + p^{k + 1} + 0}{p - 1}$

Step 3.6

Add $- 1 + p^{k + 1}$ and $0$ .

$\frac{- 1 + p^{k + 1}}{p - 1}$

Step 4

Simplify with factoring out.

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Step 4.1

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) + p^{k + 1}}{p - 1}$

Step 4.2

Factor $- 1$ out of $p^{k + 1}$ .

$\frac{- 1 (1) - 1 (- p^{k + 1})}{p - 1}$

Step 4.3

Factor $- 1$ out of $- 1 (1) - 1 (- p^{k + 1})$ .

$\frac{- 1 (1 - p^{k + 1})}{p - 1}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1 - p^{k + 1}}{p - 1}$