Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.2

One to any power is one.

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

Step 1.3

Raise $2$ to the power of $2$ .

$\frac{1}{4} + 2 \cdot \frac{1}{2} \cdot \frac{1}{3} + {(\frac{1}{3})}^{2}$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1}{4} + 2 \cdot \frac{1}{2} \cdot \frac{1}{3} + {(\frac{1}{3})}^{2}$

Step 1.4.2

Rewrite the expression.

$\frac{1}{4} + 1 \cdot \frac{1}{3} + {(\frac{1}{3})}^{2}$

Step 1.5

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

$\frac{1}{4} + \frac{1}{3} + {(\frac{1}{3})}^{2}$

Step 1.6

Apply the product rule to $\frac{1}{3}$ .

$\frac{1}{4} + \frac{1}{3} + \frac{1^{2}}{3^{2}}$

Step 1.7

One to any power is one.

$\frac{1}{4} + \frac{1}{3} + \frac{1}{3^{2}}$

Step 1.8

Raise $3$ to the power of $2$ .

$\frac{1}{4} + \frac{1}{3} + \frac{1}{9}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\frac{1}{4} \cdot \frac{9}{9} + \frac{1}{3} + \frac{1}{9}$

Step 2.2

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\frac{9}{4 \cdot 9} + \frac{1}{3} + \frac{1}{9}$

Step 2.3

Multiply $\frac{1}{3}$ by $\frac{12}{12}$ .

$\frac{9}{4 \cdot 9} + \frac{1}{3} \cdot \frac{12}{12} + \frac{1}{9}$

Step 2.4

Multiply $\frac{1}{3}$ by $\frac{12}{12}$ .

$\frac{9}{4 \cdot 9} + \frac{12}{3 \cdot 12} + \frac{1}{9}$

Step 2.5

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\frac{9}{4 \cdot 9} + \frac{12}{3 \cdot 12} + \frac{1}{9} \cdot \frac{4}{4}$

Step 2.6

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\frac{9}{4 \cdot 9} + \frac{12}{3 \cdot 12} + \frac{4}{9 \cdot 4}$

Step 2.7

Multiply $4$ by $9$ .

$\frac{9}{36} + \frac{12}{3 \cdot 12} + \frac{4}{9 \cdot 4}$

Step 2.8

Multiply $3$ by $12$ .

$\frac{9}{36} + \frac{12}{36} + \frac{4}{9 \cdot 4}$

Step 2.9

Reorder the factors of $9 \cdot 4$ .

$\frac{9}{36} + \frac{12}{36} + \frac{4}{4 \cdot 9}$

Step 2.10

Multiply $4$ by $9$ .

$\frac{9}{36} + \frac{12}{36} + \frac{4}{36}$

Step 3

Combine the numerators over the common denominator.

$\frac{9 + 12 + 4}{36}$

Step 4

Simplify by adding numbers.

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Step 4.1

Add $9$ and $12$ .

$\frac{21 + 4}{36}$

Step 4.2

Add $21$ and $4$ .

$\frac{25}{36}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{25}{36}$

Decimal Form:

$0.69\overline{4}$