Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(2 + 3 \sqrt{3}) (\sqrt{3} - 5)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot - 5 + 5 \sqrt{3} - 25}$

Step 2.3

Simplify.

$\frac{(2 + 3 \sqrt{3}) (\sqrt{3} - 5)}{- 22}$

Step 3

Expand $(2 + 3 \sqrt{3}) (\sqrt{3} - 5)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{2 (\sqrt{3} - 5) + 3 \sqrt{3} (\sqrt{3} - 5)}{- 22}$

Step 3.2

Apply the distributive property.

$\frac{2 \sqrt{3} + 2 \cdot - 5 + 3 \sqrt{3} (\sqrt{3} - 5)}{- 22}$

Step 3.3

Apply the distributive property.

$\frac{2 \sqrt{3} + 2 \cdot - 5 + 3 \sqrt{3} \sqrt{3} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $2$ by $- 5$ .

$\frac{2 \sqrt{3} - 10 + 3 \sqrt{3} \sqrt{3} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.2

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

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Step 4.1.2.1

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

$\frac{2 \sqrt{3} - 10 + 3 ({\sqrt{3}}^{1} \sqrt{3}) + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.2.2

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

$\frac{2 \sqrt{3} - 10 + 3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.2.3

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

$\frac{2 \sqrt{3} - 10 + 3 {\sqrt{3}}^{1 + 1} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{2 \sqrt{3} - 10 + 3 {\sqrt{3}}^{2} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{2 \sqrt{3} - 10 + 3 {(3^{\frac{1}{2}})}^{2} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.3.2

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

$\frac{2 \sqrt{3} - 10 + 3 \cdot 3^{\frac{1}{2} \cdot 2} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.3.3

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

$\frac{2 \sqrt{3} - 10 + 3 \cdot 3^{\frac{2}{2}} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{2 \sqrt{3} - 10 + 3 \cdot 3^{\frac{2}{2}} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{2 \sqrt{3} - 10 + 3 \cdot 3^{1} + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{2 \sqrt{3} - 10 + 3 \cdot 3 + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.4

Multiply $3$ by $3$ .

$\frac{2 \sqrt{3} - 10 + 9 + 3 \sqrt{3} \cdot - 5}{- 22}$

Step 4.1.5

Multiply $- 5$ by $3$ .

$\frac{2 \sqrt{3} - 10 + 9 - 15 \sqrt{3}}{- 22}$

Step 4.2

Subtract $15 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{- 10 + 9 - 13 \sqrt{3}}{- 22}$

Step 4.3

Add $- 10$ and $9$ .

$\frac{- 1 - 13 \sqrt{3}}{- 22}$

Step 5

Move the negative in front of the fraction.

$- \frac{- 1 - 13 \sqrt{3}}{22}$

Step 6

Rewrite $- 1$ as $- 1 (1)$ .

$- \frac{- 1 (1) - 13 \sqrt{3}}{22}$

Step 7

Factor $- 1$ out of $- 13 \sqrt{3}$ .

$- \frac{- 1 (1) - (13 \sqrt{3})}{22}$

Step 8

Factor $- 1$ out of $- 1 (1) - (13 \sqrt{3})$ .

$- \frac{- 1 (1 + 13 \sqrt{3})}{22}$

Step 9

Simplify the expression.

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Step 9.1

Move the negative in front of the fraction.

$- - \frac{1 + 13 \sqrt{3}}{22}$

Step 9.2

Multiply $- 1$ by $- 1$ .

$1 \frac{1 + 13 \sqrt{3}}{22}$

Step 9.3

Multiply $\frac{1 + 13 \sqrt{3}}{22}$ by $1$ .

$\frac{1 + 13 \sqrt{3}}{22}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1 + 13 \sqrt{3}}{22}$

Decimal Form:

$1.06893911 \dots$