Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $18$ and $81$ .

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Step 1.1.1

Factor $9$ out of $18$ .

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

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $9$ out of $81$ .

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

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

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

Step 1.2

Rewrite $\sqrt{\frac{2}{9}}$ as $\frac{\sqrt{2}}{\sqrt{9}}$ .

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{2}}{\sqrt{3^{2}}} + \sqrt{\frac{18}{9}}$

Step 1.3.2

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

$\frac{\sqrt{2}}{3} + \sqrt{\frac{18}{9}}$

Step 1.4

Divide $18$ by $9$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

Combine $\sqrt{2}$ and $\frac{3}{3}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Move $3$ to the left of $\sqrt{2}$ .

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

Step 4.2

Add $\sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{4 \sqrt{2}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \sqrt{2}}{3}$

Decimal Form:

$1.88561808 \dots$