Basic Math Examples

$5 \sqrt{\frac{1}{3}} + \sqrt{3}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$5 \frac{\sqrt{1}}{\sqrt{3}} + \sqrt{3}$

Step 1.2

Any root of $1$ is $1$ .

$5 \frac{1}{\sqrt{3}} + \sqrt{3}$

Step 1.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$5 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + \sqrt{3}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$5 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} + \sqrt{3}$

Step 1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + \sqrt{3}$

Step 1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + \sqrt{3}$

Step 1.4.4

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

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} + \sqrt{3}$

Step 1.4.5

Add $1$ and $1$ .

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{2}} + \sqrt{3}$

Step 1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$5 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + \sqrt{3}$

Step 1.4.6.2

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

$5 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \sqrt{3}$

Step 1.4.6.3

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

$5 \frac{\sqrt{3}}{3^{\frac{2}{2}}} + \sqrt{3}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$5 \frac{\sqrt{3}}{3^{\frac{2}{2}}} + \sqrt{3}$

Step 1.4.6.4.2

Rewrite the expression.

$5 \frac{\sqrt{3}}{3^{1}} + \sqrt{3}$

Step 1.4.6.5

Evaluate the exponent.

$5 \frac{\sqrt{3}}{3} + \sqrt{3}$

Step 1.5

Combine $5$ and $\frac{\sqrt{3}}{3}$ .

$\frac{5 \sqrt{3}}{3} + \sqrt{3}$

Step 2

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

$\frac{5 \sqrt{3}}{3} + \sqrt{3} \cdot \frac{3}{3}$

Step 3

Combine fractions.

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Step 3.1

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

$\frac{5 \sqrt{3}}{3} + \frac{\sqrt{3} \cdot 3}{3}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 \sqrt{3} + \sqrt{3} \cdot 3}{3}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{5 \sqrt{3} + 3 \cdot \sqrt{3}}{3}$

Step 4.2

Add $5 \sqrt{3}$ and $3 \sqrt{3}$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$4.61880215 \dots$