Trigonometry Examples

$\frac{5 \sqrt{3} - 2}{\sqrt{3} - 1}$

Step 1

Multiply $\frac{5 \sqrt{3} - 2}{\sqrt{3} - 1}$ by $\frac{\sqrt{3} + 1}{\sqrt{3} + 1}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 \sqrt{3} - 2}{\sqrt{3} - 1}$ by $\frac{\sqrt{3} + 1}{\sqrt{3} + 1}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(5 \sqrt{3} - 2) (\sqrt{3} + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $5 \sqrt{3} \sqrt{3}$ .

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

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

Step 4.1.2.4.2

Rewrite the expression.

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

Step 4.1.2.5

Evaluate the exponent.

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

Step 4.1.3

Multiply $5$ by $3$ .

$\frac{15 + 5 \sqrt{3} \cdot 1 - 2 \sqrt{3} - 2 \cdot 1}{2}$

Step 4.1.4

Multiply $5$ by $1$ .

$\frac{15 + 5 \sqrt{3} - 2 \sqrt{3} - 2 \cdot 1}{2}$

Step 4.1.5

Multiply $- 2$ by $1$ .

$\frac{15 + 5 \sqrt{3} - 2 \sqrt{3} - 2}{2}$

Step 4.2

Subtract $2$ from $15$ .

$\frac{13 + 5 \sqrt{3} - 2 \sqrt{3}}{2}$

Step 4.3

Subtract $2 \sqrt{3}$ from $5 \sqrt{3}$ .

$\frac{13 + 3 \sqrt{3}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{13 + 3 \sqrt{3}}{2}$

Decimal Form:

$9.09807621 \dots$