Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

$\frac{5 (\sqrt{3} + 2)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot 2 - 2 \sqrt{3} - 4} - \frac{2}{\sqrt{3}}$

Step 1.4

Simplify.

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

Step 1.5

Move the negative one from the denominator of $\frac{5 (\sqrt{3} + 2)}{- 1}$ .

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

Step 1.6

Rewrite $- 1 \cdot (5 (\sqrt{3} + 2))$ as $- (5 (\sqrt{3} + 2))$ .

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

Step 1.7

Apply the distributive property.

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

Step 1.8

Multiply $5$ by $2$ .

$- (5 \sqrt{3} + 10) - \frac{2}{\sqrt{3}}$

Step 1.9

Apply the distributive property.

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

Step 1.10

Multiply $5$ by $- 1$ .

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

Step 1.11

Multiply $- 1$ by $10$ .

$- 5 \sqrt{3} - 10 - \frac{2}{\sqrt{3}}$

Step 1.12

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

$- 5 \sqrt{3} - 10 - (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.13

Combine and simplify the denominator.

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Step 1.13.1

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

$- 5 \sqrt{3} - 10 - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.13.2

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

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

Step 1.13.3

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

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

Step 1.13.4

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

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

Step 1.13.5

Add $1$ and $1$ .

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

Step 1.13.6

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

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Step 1.13.6.1

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

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

Step 1.13.6.2

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

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

Step 1.13.6.3

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

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

Step 1.13.6.4

Cancel the common factor of $2$ .

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Step 1.13.6.4.1

Cancel the common factor.

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

Step 1.13.6.4.2

Rewrite the expression.

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

Step 1.13.6.5

Evaluate the exponent.

$- 5 \sqrt{3} - 10 - \frac{2 \sqrt{3}}{3}$

Step 2

Find the common denominator.

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Step 2.1

Write $- 5 \sqrt{3}$ as a fraction with denominator $1$ .

$\frac{- 5 \sqrt{3}}{1} - 10 - \frac{2 \sqrt{3}}{3}$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Write $- 10$ as a fraction with denominator $1$ .

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

Step 2.5

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

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

Step 2.6

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $- 5$ .

$\frac{- 15 \sqrt{3} - 10 \cdot 3 - 2 \sqrt{3}}{3}$

Step 4.2

Multiply $- 10$ by $3$ .

$\frac{- 15 \sqrt{3} - 30 - 2 \sqrt{3}}{3}$

Step 5

Simplify terms.

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Step 5.1

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

$\frac{- 30 - 17 \sqrt{3}}{3}$

Step 5.2

Rewrite $- 30$ as $- 1 (30)$ .

$\frac{- 1 (30) - 17 \sqrt{3}}{3}$

Step 5.3

Factor $- 1$ out of $- 17 \sqrt{3}$ .

$\frac{- 1 (30) - (17 \sqrt{3})}{3}$

Step 5.4

Factor $- 1$ out of $- 1 (30) - (17 \sqrt{3})$ .

$\frac{- 1 (30 + 17 \sqrt{3})}{3}$

Step 5.5

Move the negative in front of the fraction.

$- \frac{30 + 17 \sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{30 + 17 \sqrt{3}}{3}$

Decimal Form:

$- 19.81495457 \dots$