Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.1.1

Factor $4$ out of $8$ .

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

Step 1.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.2

Pull terms out from under the radical.

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $81$ as $9^{2}$ .

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

Step 1.3.2

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

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

Step 1.4

Divide $18$ by $9$ .

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

Step 2

To write $\sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 3

Combine fractions.

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Step 3.1

Combine $\sqrt{2}$ and $\frac{9}{9}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

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:

$1.72848324 \dots$