Pre-Algebra Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

\sqrt{2}

$\sqrt{2}$ .

\frac{\sqrt{2}}{2} + \frac{1}{3} \cdot \sqrt{2}

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

Step 1.2

Combine

\frac{1}{3}

$\frac{1}{3}$ and

\sqrt{2}

$\sqrt{2}$ .

\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{3}

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

\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{3}

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

Step 2

To write

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{\sqrt{2}}{2} \cdot \frac{3}{3} + \frac{\sqrt{2}}{3}

$\frac{\sqrt{2}}{2} \cdot \frac{3}{3} + \frac{\sqrt{2}}{3}$

Step 3

To write

\frac{\sqrt{2}}{3}

$\frac{\sqrt{2}}{3}$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{\sqrt{2}}{2} \cdot \frac{3}{3} + \frac{\sqrt{2}}{3} \cdot \frac{2}{2}

$\frac{\sqrt{2}}{2} \cdot \frac{3}{3} + \frac{\sqrt{2}}{3} \cdot \frac{2}{2}$

Step 4

Write each expression with a common denominator of

6

$6$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 4.1

Multiply

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{\sqrt{2} \cdot 3}{2 \cdot 3} + \frac{\sqrt{2}}{3} \cdot \frac{2}{2}

$\frac{\sqrt{2} \cdot 3}{2 \cdot 3} + \frac{\sqrt{2}}{3} \cdot \frac{2}{2}$

Step 4.2

Multiply

2

$2$ by

3

$3$ .

\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2}}{3} \cdot \frac{2}{2}

$\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2}}{3} \cdot \frac{2}{2}$

Step 4.3

Multiply

\frac{\sqrt{2}}{3}

$\frac{\sqrt{2}}{3}$ by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2} \cdot 2}{3 \cdot 2}

$\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2} \cdot 2}{3 \cdot 2}$

Step 4.4

Multiply

3

$3$ by

2

$2$ .

\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2} \cdot 2}{6}

$\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2} \cdot 2}{6}$

\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2} \cdot 2}{6}

$\frac{\sqrt{2} \cdot 3}{6} + \frac{\sqrt{2} \cdot 2}{6}$

Step 5

Combine the numerators over the common denominator.

\frac{\sqrt{2} \cdot 3 + \sqrt{2} \cdot 2}{6}

$\frac{\sqrt{2} \cdot 3 + \sqrt{2} \cdot 2}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Move

3

$3$ to the left of

\sqrt{2}

$\sqrt{2}$ .

\frac{3 \cdot \sqrt{2} + \sqrt{2} \cdot 2}{6}

$\frac{3 \cdot \sqrt{2} + \sqrt{2} \cdot 2}{6}$

Step 6.2

Move

2

$2$ to the left of

\sqrt{2}

$\sqrt{2}$ .

\frac{3 \sqrt{2} + 2 \cdot \sqrt{2}}{6}

$\frac{3 \sqrt{2} + 2 \cdot \sqrt{2}}{6}$

Step 6.3

Add

3 \sqrt{2}

$3 \sqrt{2}$ and

2 \sqrt{2}

$2 \sqrt{2}$ .

\frac{5 \sqrt{2}}{6}

$\frac{5 \sqrt{2}}{6}$

\frac{5 \sqrt{2}}{6}

$\frac{5 \sqrt{2}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

\frac{5 \sqrt{2}}{6}

$\frac{5 \sqrt{2}}{6}$

Decimal Form:

1.17851130 \dots

$1.17851130 \dots$