Algebra Examples

$\sqrt{\frac{8}{16}} + \sqrt{\frac{8}{4}}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $8$ and $16$ .

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Step 1.1.1

Factor $8$ out of $8$ .

$\sqrt{\frac{8 (1)}{16}} + \sqrt{\frac{8}{4}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $8$ out of $16$ .

$\sqrt{\frac{8 \cdot 1}{8 \cdot 2}} + \sqrt{\frac{8}{4}}$

Step 1.1.2.2

Cancel the common factor.

$\sqrt{\frac{8 \cdot 1}{8 \cdot 2}} + \sqrt{\frac{8}{4}}$

Step 1.1.2.3

Rewrite the expression.

$\sqrt{\frac{1}{2}} + \sqrt{\frac{8}{4}}$

Step 1.2

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

$\frac{\sqrt{1}}{\sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.3

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.4

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

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.5.2

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

$\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.5.3

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

$\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \sqrt{\frac{8}{4}}$

Step 1.5.4

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

$\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \sqrt{\frac{8}{4}}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{2}} + \sqrt{\frac{8}{4}}$

Step 1.5.6

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

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Step 1.5.6.1

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

$\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \sqrt{\frac{8}{4}}$

Step 1.5.6.2

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

$\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \sqrt{\frac{8}{4}}$

Step 1.5.6.3

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

$\frac{\sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{\frac{8}{4}}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{\frac{8}{4}}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{2^{1}} + \sqrt{\frac{8}{4}}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{\sqrt{2}}{2} + \sqrt{\frac{8}{4}}$

Step 1.6

Divide $8$ by $4$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.12132034 \dots$