Basic Math Examples

\sqrt{\frac{8}{81}} + \sqrt{\frac{18}{9}}

$\sqrt{\frac{8}{81}} + \sqrt{\frac{18}{9}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite

\sqrt{\frac{8}{81}}

$\sqrt{\frac{8}{81}}$ as

\frac{\sqrt{8}}{\sqrt{81}}

$\frac{\sqrt{8}}{\sqrt{81}}$ .

\frac{\sqrt{8}}{\sqrt{81}} + \sqrt{\frac{18}{9}}

$\frac{\sqrt{8}}{\sqrt{81}} + \sqrt{\frac{18}{9}}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite

8

$8$ as

2^{2} \cdot 2

$2^{2} \cdot 2$ .

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Step 1.2.1.1

Factor

4

$4$ out of

8

$8$ .

\frac{\sqrt{4 (2)}}{\sqrt{81}} + \sqrt{\frac{18}{9}}

$\frac{\sqrt{4 (2)}}{\sqrt{81}} + \sqrt{\frac{18}{9}}$

Step 1.2.1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}

$\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}$

\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}

$\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}$

Step 1.2.2

Pull terms out from under the radical.

\frac{2 \sqrt{2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}

$\frac{2 \sqrt{2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}$

\frac{2 \sqrt{2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}

$\frac{2 \sqrt{2}}{\sqrt{81}} + \sqrt{\frac{18}{9}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite

81

$81$ as

9^{2}

$9^{2}$ .

\frac{2 \sqrt{2}}{\sqrt{9^{2}}} + \sqrt{\frac{18}{9}}

$\frac{2 \sqrt{2}}{\sqrt{9^{2}}} + \sqrt{\frac{18}{9}}$

Step 1.3.2

Pull terms out from under the radical, assuming positive real numbers.

\frac{2 \sqrt{2}}{9} + \sqrt{\frac{18}{9}}

$\frac{2 \sqrt{2}}{9} + \sqrt{\frac{18}{9}}$

\frac{2 \sqrt{2}}{9} + \sqrt{\frac{18}{9}}

$\frac{2 \sqrt{2}}{9} + \sqrt{\frac{18}{9}}$

Step 1.4

Divide

18

$18$ by

9

$9$ .

\frac{2 \sqrt{2}}{9} + \sqrt{2}

$\frac{2 \sqrt{2}}{9} + \sqrt{2}$

\frac{2 \sqrt{2}}{9} + \sqrt{2}

$\frac{2 \sqrt{2}}{9} + \sqrt{2}$

Step 2

To write

\sqrt{2}

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

\frac{9}{9}

$\frac{9}{9}$ .

\frac{2 \sqrt{2}}{9} + \sqrt{2} \cdot \frac{9}{9}

$\frac{2 \sqrt{2}}{9} + \sqrt{2} \cdot \frac{9}{9}$

Step 3

Combine fractions.

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Step 3.1

Combine

\sqrt{2}

$\sqrt{2}$ and

\frac{9}{9}

$\frac{9}{9}$ .

\frac{2 \sqrt{2}}{9} + \frac{\sqrt{2} \cdot 9}{9}

$\frac{2 \sqrt{2}}{9} + \frac{\sqrt{2} \cdot 9}{9}$

Step 3.2

Combine the numerators over the common denominator.

\frac{2 \sqrt{2} + \sqrt{2} \cdot 9}{9}

$\frac{2 \sqrt{2} + \sqrt{2} \cdot 9}{9}$

\frac{2 \sqrt{2} + \sqrt{2} \cdot 9}{9}

$\frac{2 \sqrt{2} + \sqrt{2} \cdot 9}{9}$

Step 4

Simplify the numerator.

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Step 4.1

Move

$9$ to the left of

$\sqrt{2}$ .

$\frac{2 \sqrt{2} + 9 \cdot \sqrt{2}}{9}$

Step 4.2

Add

$2 \sqrt{2}$ and

$9 \sqrt{2}$ .

$\frac{11 \sqrt{2}}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{11 \sqrt{2}}{9}$

Decimal Form:

$1.72848324 \dots$