Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

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

Step 1.1.2

Combine the numerators over the common denominator.

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

Step 1.1.3

Add $2$ and $1$ .

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Rewrite $\frac{\frac{3}{2} + 1}{\frac{3}{2}}$ in a factored form.

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Step 1.2.3.1

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

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

Step 1.2.3.2

Combine the numerators over the common denominator.

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

Step 1.2.3.3

Add $3$ and $2$ .

$2 + \frac{1}{3 + \frac{1}{\frac{\frac{5}{2}}{\frac{3}{2}}}}$

Step 1.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 + \frac{1}{3 + \frac{1}{\frac{5}{2} \cdot \frac{2}{3}}}$

Step 1.2.5

Cancel the common factor of $2$ .

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Step 1.2.5.1

Cancel the common factor.

$2 + \frac{1}{3 + \frac{1}{\frac{5}{2} \cdot \frac{2}{3}}}$

Step 1.2.5.2

Rewrite the expression.

$2 + \frac{1}{3 + \frac{1}{5 (\frac{1}{3})}}$

Step 1.2.6

Combine $5$ and $\frac{1}{3}$ .

$2 + \frac{1}{3 + \frac{1}{\frac{5}{3}}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{\frac{5}{3}}{\frac{5}{3}}$ .

$2 + \frac{1}{\frac{3 (\frac{5}{3})}{\frac{5}{3}} + \frac{1}{\frac{5}{3}}}$

Step 1.3.2

Combine the numerators over the common denominator.

$2 + \frac{1}{\frac{3 (\frac{5}{3}) + 1}{\frac{5}{3}}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Divide $5$ by $3$ .

$2 + \frac{1}{\frac{3 \cdot 1.\overline{6} + 1}{\frac{5}{3}}}$

Step 1.3.3.2

Multiply $3$ by $1.\overline{6}$ .

$2 + \frac{1}{\frac{5 + 1}{\frac{5}{3}}}$

Step 1.3.3.3

Add $5$ and $1$ .

$2 + \frac{1}{\frac{6}{\frac{5}{3}}}$

Step 1.3.4

Multiply the numerator by the reciprocal of the denominator.

$2 + \frac{1}{6 (\frac{3}{5})}$

Step 1.3.5

Multiply $6 (\frac{3}{5})$ .

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Step 1.3.5.1

Combine $6$ and $\frac{3}{5}$ .

$2 + \frac{1}{\frac{6 \cdot 3}{5}}$

Step 1.3.5.2

Multiply $6$ by $3$ .

$2 + \frac{1}{\frac{18}{5}}$

Step 1.3.6

Cancel the common factor of $18$ and $5$ .

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Step 1.3.6.1

Rewrite $18$ as $1 (18)$ .

$2 + \frac{1}{\frac{1 (18)}{5}}$

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

Rewrite $5$ as $1 (5)$ .

$2 + \frac{1}{\frac{1 \cdot 18}{1 \cdot 5}}$

Step 1.3.6.2.2

Cancel the common factor.

$2 + \frac{1}{\frac{1 \cdot 18}{1 \cdot 5}}$

Step 1.3.6.2.3

Rewrite the expression.

$2 + \frac{1}{\frac{18}{5}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$2 + 1 (\frac{5}{18})$

Step 1.5

Multiply $\frac{5}{18}$ by $1$ .

$2 + \frac{5}{18}$

Step 2

To write $2$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$2 \cdot \frac{18}{18} + \frac{5}{18}$

Step 3

Combine $2$ and $\frac{18}{18}$ .

$\frac{2 \cdot 18}{18} + \frac{5}{18}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 \cdot 18 + 5}{18}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $18$ .

$\frac{36 + 5}{18}$

Step 5.2

Add $36$ and $5$ .

$\frac{41}{18}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{41}{18}$

Decimal Form:

$2.2\overline{7}$

Mixed Number Form:

$2 \frac{5}{18}$