Precalculus Examples

$1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27}$

Step 1

Find the common denominator.

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Step 1.1

Write $1$ as a fraction with denominator $1$ .

$\frac{1}{1} + \frac{1}{3} + \frac{1}{9} + \frac{1}{27}$

Step 1.2

Multiply $\frac{1}{1}$ by $\frac{27}{27}$ .

$\frac{1}{1} \cdot \frac{27}{27} + \frac{1}{3} + \frac{1}{9} + \frac{1}{27}$

Step 1.3

Multiply $\frac{1}{1}$ by $\frac{27}{27}$ .

$\frac{27}{27} + \frac{1}{3} + \frac{1}{9} + \frac{1}{27}$

Step 1.4

Multiply $\frac{1}{3}$ by $\frac{9}{9}$ .

$\frac{27}{27} + \frac{1}{3} \cdot \frac{9}{9} + \frac{1}{9} + \frac{1}{27}$

Step 1.5

Multiply $\frac{1}{3}$ by $\frac{9}{9}$ .

$\frac{27}{27} + \frac{9}{3 \cdot 9} + \frac{1}{9} + \frac{1}{27}$

Step 1.6

Multiply $\frac{1}{9}$ by $\frac{3}{3}$ .

$\frac{27}{27} + \frac{9}{3 \cdot 9} + \frac{1}{9} \cdot \frac{3}{3} + \frac{1}{27}$

Step 1.7

Multiply $\frac{1}{9}$ by $\frac{3}{3}$ .

$\frac{27}{27} + \frac{9}{3 \cdot 9} + \frac{3}{9 \cdot 3} + \frac{1}{27}$

Step 1.8

Multiply $3$ by $9$ .

$\frac{27}{27} + \frac{9}{27} + \frac{3}{9 \cdot 3} + \frac{1}{27}$

Step 1.9

Reorder the factors of $9 \cdot 3$ .

$\frac{27}{27} + \frac{9}{27} + \frac{3}{3 \cdot 9} + \frac{1}{27}$

Step 1.10

Multiply $3$ by $9$ .

$\frac{27}{27} + \frac{9}{27} + \frac{3}{27} + \frac{1}{27}$

Step 2

Combine the numerators over the common denominator.

$\frac{27 + 9 + 3 + 1}{27}$

Step 3

Simplify by adding numbers.

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Step 3.1

Add $27$ and $9$ .

$\frac{36 + 3 + 1}{27}$

Step 3.2

Add $36$ and $3$ .

$\frac{39 + 1}{27}$

Step 3.3

Add $39$ and $1$ .

$\frac{40}{27}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{40}{27}$

Decimal Form:

$1.\overline{481}$

Mixed Number Form:

$1 \frac{13}{27}$