Algebra Examples

$\sqrt{\frac{18}{16}} + \sqrt{\frac{8}{4}}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $18$ and $16$ .

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Step 1.1.1

Factor $2$ out of $18$ .

$\sqrt{\frac{2 (9)}{16}} + \sqrt{\frac{8}{4}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $16$ .

$\sqrt{\frac{2 \cdot 9}{2 \cdot 8}} + \sqrt{\frac{8}{4}}$

Step 1.1.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 9}{2 \cdot 8}} + \sqrt{\frac{8}{4}}$

Step 1.1.2.3

Rewrite the expression.

$\sqrt{\frac{9}{8}} + \sqrt{\frac{8}{4}}$

Step 1.2

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

$\frac{\sqrt{9}}{\sqrt{8}} + \sqrt{\frac{8}{4}}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

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

$\frac{\sqrt{3^{2}}}{\sqrt{8}} + \sqrt{\frac{8}{4}}$

Step 1.3.2

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

$\frac{3}{\sqrt{8}} + \sqrt{\frac{8}{4}}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

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

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Step 1.4.1.1

Factor $4$ out of $8$ .

$\frac{3}{\sqrt{4 (2)}} + \sqrt{\frac{8}{4}}$

Step 1.4.1.2

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

$\frac{3}{\sqrt{2^{2} \cdot 2}} + \sqrt{\frac{8}{4}}$

Step 1.4.2

Pull terms out from under the radical.

$\frac{3}{2 \sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.5

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

$\frac{3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

$\frac{3 \sqrt{2}}{2 \sqrt{2} \sqrt{2}} + \sqrt{\frac{8}{4}}$

Step 1.6.2

Move $\sqrt{2}$ .

$\frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})} + \sqrt{\frac{8}{4}}$

Step 1.6.3

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

$\frac{3 \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})} + \sqrt{\frac{8}{4}}$

Step 1.6.4

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

$\frac{3 \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} + \sqrt{\frac{8}{4}}$

Step 1.6.5

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

$\frac{3 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}} + \sqrt{\frac{8}{4}}$

Step 1.6.6

Add $1$ and $1$ .

$\frac{3 \sqrt{2}}{2 {\sqrt{2}}^{2}} + \sqrt{\frac{8}{4}}$

Step 1.6.7

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

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Step 1.6.7.1

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

$\frac{3 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}} + \sqrt{\frac{8}{4}}$

Step 1.6.7.2

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

$\frac{3 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}} + \sqrt{\frac{8}{4}}$

Step 1.6.7.3

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

$\frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}} + \sqrt{\frac{8}{4}}$

Step 1.6.7.4

Cancel the common factor of $2$ .

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Step 1.6.7.4.1

Cancel the common factor.

$\frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}} + \sqrt{\frac{8}{4}}$

Step 1.6.7.4.2

Rewrite the expression.

$\frac{3 \sqrt{2}}{2 \cdot 2^{1}} + \sqrt{\frac{8}{4}}$

Step 1.6.7.5

Evaluate the exponent.

$\frac{3 \sqrt{2}}{2 \cdot 2} + \sqrt{\frac{8}{4}}$

Step 1.7

Multiply $2$ by $2$ .

$\frac{3 \sqrt{2}}{4} + \sqrt{\frac{8}{4}}$

Step 1.8

Divide $8$ by $4$ .

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

Step 2

To write $\sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3

Combine fractions.

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Step 3.1

Combine $\sqrt{2}$ and $\frac{4}{4}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Move $4$ to the left of $\sqrt{2}$ .

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

Step 4.2

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

$\frac{7 \sqrt{2}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{2}}{4}$

Decimal Form:

$2.47487373 \dots$