Algebra Examples

$\frac{3 + \sqrt{3}}{4 \sqrt{12}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 1.4

Multiply $4$ by $2$ .

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

Step 2

Multiply $\frac{3 + \sqrt{3}}{8 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{3 + \sqrt{3}}{8 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2

Move $\sqrt{3}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Evaluate the exponent.

$\frac{(3 + \sqrt{3}) \sqrt{3}}{8 \cdot 3}$

Step 4

Multiply $8$ by $3$ .

$\frac{(3 + \sqrt{3}) \sqrt{3}}{24}$

Step 5

Apply the distributive property.

$\frac{3 \sqrt{3} + \sqrt{3} \sqrt{3}}{24}$

Step 6

Combine using the product rule for radicals.

$\frac{3 \sqrt{3} + \sqrt{3 \cdot 3}}{24}$

Step 7

Simplify each term.

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Step 7.1

Multiply $3$ by $3$ .

$\frac{3 \sqrt{3} + \sqrt{9}}{24}$

Step 7.2

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

$\frac{3 \sqrt{3} + \sqrt{3^{2}}}{24}$

Step 7.3

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

$\frac{3 \sqrt{3} + 3}{24}$

Step 8

Cancel the common factor of $3 \sqrt{3} + 3$ and $24$ .

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Step 8.1

Factor $3$ out of $3 \sqrt{3}$ .

$\frac{3 (\sqrt{3}) + 3}{24}$

Step 8.2

Factor $3$ out of $3$ .

$\frac{3 (\sqrt{3}) + 3 (1)}{24}$

Step 8.3

Factor $3$ out of $3 (\sqrt{3}) + 3 (1)$ .

$\frac{3 (\sqrt{3} + 1)}{24}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $3$ out of $24$ .

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

Step 8.4.2

Cancel the common factor.

$\frac{3 (\sqrt{3} + 1)}{3 \cdot 8}$

Step 8.4.3

Rewrite the expression.

$\frac{\sqrt{3} + 1}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3} + 1}{8}$

Decimal Form:

$0.34150635 \dots$