Pre-Algebra Examples

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

Step 1

To find the reciprocal, divide $1$ by the number given.

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

Step 2

Simplify.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

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

Step 2.1.2

Combine the numerators over the common denominator.

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

Step 2.1.3

Subtract $3$ from $2$ .

$\frac{1}{2 - \frac{1}{2 - \frac{1}{\frac{- 1}{2}}}}$

Step 2.1.4

Move the negative in front of the fraction.

$\frac{1}{2 - \frac{1}{2 - \frac{1}{- \frac{1}{2}}}}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Cancel the common factor of $1$ and $- 1$ .

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Step 2.2.1.1

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

$\frac{1}{2 - \frac{1}{2 - \frac{- 1 (- 1)}{- \frac{1}{2}}}}$

Step 2.2.1.2

Move the negative in front of the fraction.

$\frac{1}{2 - \frac{1}{2 - - \frac{1}{\frac{1}{2}}}}$

Step 2.2.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2 - \frac{1}{2 - - (1 \cdot 2)}}$

Step 2.2.3

Multiply $- - (1 \cdot 2)$ .

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Step 2.2.3.1

Multiply $2$ by $1$ .

$\frac{1}{2 - \frac{1}{2 - (- 1 \cdot 2)}}$

Step 2.2.3.2

Multiply $- 1$ by $2$ .

$\frac{1}{2 - \frac{1}{2 - - 2}}$

Step 2.2.3.3

Multiply $- 1$ by $- 2$ .

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

Step 2.2.4

Add $2$ and $2$ .

$\frac{1}{2 - \frac{1}{4}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{2 \cdot \frac{4}{4} - \frac{1}{4}}$

Step 2.3.2

Combine $2$ and $\frac{4}{4}$ .

$\frac{1}{\frac{2 \cdot 4}{4} - \frac{1}{4}}$

Step 2.3.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{2 \cdot 4 - 1}{4}}$

Step 2.3.4

Simplify the numerator.

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Step 2.3.4.1

Multiply $2$ by $4$ .

$\frac{1}{\frac{8 - 1}{4}}$

Step 2.3.4.2

Subtract $1$ from $8$ .

$\frac{1}{\frac{7}{4}}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{4}{7})$

Step 2.5

Multiply $\frac{4}{7}$ by $1$ .

$\frac{4}{7}$