Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Evaluate the exponent.

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

Step 1.3

Multiply $\frac{\sqrt{3}}{3} \cdot \frac{\sqrt{5}}{2}$ .

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Step 1.3.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{\sqrt{5}}{2}$ .

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

Step 1.3.2

Combine using the product rule for radicals.

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

Step 1.3.3

Multiply $3$ by $5$ .

$\frac{\sqrt{15}}{3 \cdot 2} + \frac{1}{\sqrt{3}} \cdot (\frac{1}{2})$

Step 1.3.4

Multiply $3$ by $2$ .

$\frac{\sqrt{15}}{6} + \frac{1}{\sqrt{3}} \cdot (\frac{1}{2})$

Step 1.4

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

$\frac{\sqrt{15}}{6} + \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot \frac{1}{2}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \frac{1}{2}$

Step 1.5.2

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot \frac{1}{2}$

Step 1.5.3

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot \frac{1}{2}$

Step 1.5.4

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot \frac{1}{2}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{{\sqrt{3}}^{2}} \cdot \frac{1}{2}$

Step 1.5.6

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

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Step 1.5.6.1

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot \frac{1}{2}$

Step 1.5.6.2

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{2}$

Step 1.5.6.3

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

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot \frac{1}{2}$

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.

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot \frac{1}{2}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{3^{1}} \cdot \frac{1}{2}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{3} \cdot \frac{1}{2}$

Step 1.6

Multiply $\frac{\sqrt{3}}{3} \cdot \frac{1}{2}$ .

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Step 1.6.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{1}{2}$ .

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{3 \cdot 2}$

Step 1.6.2

Multiply $3$ by $2$ .

$\frac{\sqrt{15}}{6} + \frac{\sqrt{3}}{6}$

Step 2

Combine the numerators over the common denominator.

$\frac{\sqrt{15} + \sqrt{3}}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{15} + \sqrt{3}}{6}$

Decimal Form:

$0.93417235 \dots$