Algebra Examples

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

Step 1

Multiply $\frac{\sqrt{3}}{2 + 4 \sqrt{3}}$ by $\frac{2 - 4 \sqrt{3}}{2 - 4 \sqrt{3}}$ .

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

Step 2

Multiply $\frac{\sqrt{3}}{2 + 4 \sqrt{3}}$ by $\frac{2 - 4 \sqrt{3}}{2 - 4 \sqrt{3}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{3} (2 - 4 \sqrt{3})}{4 - 8 \sqrt{3} + 8 \sqrt{3} - 16 {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{3} (2 - 4 \sqrt{3})}{- 44}$

Step 5

Cancel the common factor of $2 - 4 \sqrt{3}$ and $- 44$ .

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Step 5.1

Factor $2$ out of $\sqrt{3} (2 - 4 \sqrt{3})$ .

$\frac{2 (\sqrt{3} (1 - 2 \sqrt{3}))}{- 44}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $- 44$ .

$\frac{2 (\sqrt{3} (1 - 2 \sqrt{3}))}{2 (- 22)}$

Step 5.2.2

Cancel the common factor.

$\frac{2 (\sqrt{3} (1 - 2 \sqrt{3}))}{2 \cdot - 22}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt{3} (1 - 2 \sqrt{3})}{- 22}$

Step 6

Apply the distributive property.

$\frac{\sqrt{3} \cdot 1 + \sqrt{3} (- 2 \sqrt{3})}{- 22}$

Step 7

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

$\frac{\sqrt{3} + \sqrt{3} (- 2 \sqrt{3})}{- 22}$

Step 8

Multiply $\sqrt{3} (- 2 \sqrt{3})$ .

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Step 8.1

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

$\frac{\sqrt{3} - 2 ({\sqrt{3}}^{1} \sqrt{3})}{- 22}$

Step 8.2

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

$\frac{\sqrt{3} - 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{- 22}$

Step 8.3

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

$\frac{\sqrt{3} - 2 {\sqrt{3}}^{1 + 1}}{- 22}$

Step 8.4

Add $1$ and $1$ .

$\frac{\sqrt{3} - 2 {\sqrt{3}}^{2}}{- 22}$

Step 9

Simplify each term.

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Step 9.1

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

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Step 9.1.1

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

$\frac{\sqrt{3} - 2 {(3^{\frac{1}{2}})}^{2}}{- 22}$

Step 9.1.2

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

$\frac{\sqrt{3} - 2 \cdot 3^{\frac{1}{2} \cdot 2}}{- 22}$

Step 9.1.3

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

$\frac{\sqrt{3} - 2 \cdot 3^{\frac{2}{2}}}{- 22}$

Step 9.1.4

Cancel the common factor of $2$ .

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Step 9.1.4.1

Cancel the common factor.

$\frac{\sqrt{3} - 2 \cdot 3^{\frac{2}{2}}}{- 22}$

Step 9.1.4.2

Rewrite the expression.

$\frac{\sqrt{3} - 2 \cdot 3^{1}}{- 22}$

Step 9.1.5

Evaluate the exponent.

$\frac{\sqrt{3} - 2 \cdot 3}{- 22}$

Step 9.2

Multiply $- 2$ by $3$ .

$\frac{\sqrt{3} - 6}{- 22}$

Step 10

Move the negative in front of the fraction.

$- \frac{\sqrt{3} - 6}{22}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3} - 6}{22}$

Decimal Form:

$0.19399769 \dots$