Algebra Examples

$\sqrt{\frac{8}{9}} + \sqrt{\frac{6}{12}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{8}{9}}$ as $\frac{\sqrt{8}}{\sqrt{9}}$ .

$\frac{\sqrt{8}}{\sqrt{9}} + \sqrt{\frac{6}{12}}$

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 $8$ .

$\frac{\sqrt{4 (2)}}{\sqrt{9}} + \sqrt{\frac{6}{12}}$

Step 1.2.1.2

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

$\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{9}} + \sqrt{\frac{6}{12}}$

Step 1.2.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{\sqrt{9}} + \sqrt{\frac{6}{12}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

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

$\frac{2 \sqrt{2}}{\sqrt{3^{2}}} + \sqrt{\frac{6}{12}}$

Step 1.3.2

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

$\frac{2 \sqrt{2}}{3} + \sqrt{\frac{6}{12}}$

Step 1.4

Cancel the common factor of $6$ and $12$ .

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Step 1.4.1

Factor $6$ out of $6$ .

$\frac{2 \sqrt{2}}{3} + \sqrt{\frac{6 (1)}{12}}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $6$ out of $12$ .

$\frac{2 \sqrt{2}}{3} + \sqrt{\frac{6 \cdot 1}{6 \cdot 2}}$

Step 1.4.2.2

Cancel the common factor.

$\frac{2 \sqrt{2}}{3} + \sqrt{\frac{6 \cdot 1}{6 \cdot 2}}$

Step 1.4.2.3

Rewrite the expression.

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

Step 1.5

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 1.6

Any root of $1$ is $1$ .

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

Step 1.7

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.8

Combine and simplify the denominator.

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Step 1.8.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.8.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.8.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.8.5

Add $1$ and $1$ .

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

Step 1.8.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{2 \sqrt{2}}{3} + \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.8.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.8.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.8.6.4

Cancel the common factor of $2$ .

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Step 1.8.6.4.1

Cancel the common factor.

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

Step 1.8.6.4.2

Rewrite the expression.

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

Step 1.8.6.5

Evaluate the exponent.

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $3$ by $2$ .

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

Step 4.3

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

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

Step 4.4

Multiply $2$ by $3$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $2$ by $2$ .

$\frac{4 \sqrt{2} + \sqrt{2} \cdot 3}{6}$

Step 6.2

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

$\frac{4 \sqrt{2} + 3 \cdot \sqrt{2}}{6}$

Step 6.3

Add $4 \sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{7 \sqrt{2}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{2}}{6}$

Decimal Form:

$1.64991582 \dots$