Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

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

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

Step 2.1.2

Move $\sqrt{3}$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

Add $1$ and $1$ .

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

Step 2.1.7

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

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

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

Step 2.1.7.3

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

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

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

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

Step 2.1.7.4.2

Rewrite the expression.

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

Step 2.1.7.5

Evaluate the exponent.

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

Step 2.2

Multiply $2$ by $3$ .

$\frac{(\sqrt{3} + 5) \sqrt{3}}{6}$

Step 2.3

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{3} + 5 \sqrt{3}}{6}$

Step 2.4

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 3} + 5 \sqrt{3}}{6}$

Step 3

Simplify each term.

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Step 3.1

Multiply $3$ by $3$ .

$\frac{\sqrt{9} + 5 \sqrt{3}}{6}$

Step 3.2

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

$\frac{\sqrt{3^{2}} + 5 \sqrt{3}}{6}$

Step 3.3

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

$\frac{3 + 5 \sqrt{3}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 + 5 \sqrt{3}}{6}$

Decimal Form:

$1.94337567 \dots$