Finite Math Examples

$\frac{{(- \sqrt{3} + 2 \sqrt{2})}^{2} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite ${(- \sqrt{3} + 2 \sqrt{2})}^{2}$ as $(- \sqrt{3} + 2 \sqrt{2}) (- \sqrt{3} + 2 \sqrt{2})$ .

$\frac{(- \sqrt{3} + 2 \sqrt{2}) (- \sqrt{3} + 2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\frac{- \sqrt{3} (- \sqrt{3} + 2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3} + 2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.2.2

Apply the distributive property.

$\frac{- \sqrt{3} (- \sqrt{3}) - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3} + 2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.2.3

Apply the distributive property.

$\frac{- \sqrt{3} (- \sqrt{3}) - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

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

Tap for more steps...

Step 1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{1 \sqrt{3} \sqrt{3} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.1.2

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

$\frac{\sqrt{3} \sqrt{3} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.1.3

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

$\frac{{\sqrt{3}}^{1} \sqrt{3} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.1.4

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

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.1.5

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

$\frac{{\sqrt{3}}^{1 + 1} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.1.6

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.2

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

Tap for more steps...

Step 1.3.1.2.1

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

$\frac{{(3^{\frac{1}{2}})}^{2} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.2.2

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

$\frac{3^{\frac{1}{2} \cdot 2} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.2.3

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

$\frac{3^{\frac{2}{2}} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.1.2.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.2.4.2

Rewrite the expression.

$\frac{3^{1} - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.2.5

Evaluate the exponent.

$\frac{3 - \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.3

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

Tap for more steps...

Step 1.3.1.3.1

Multiply $2$ by $- 1$ .

$\frac{3 - 2 \sqrt{3} \sqrt{2} + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.3.2

Combine using the product rule for radicals.

$\frac{3 - 2 \sqrt{2 \cdot 3} + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.3.3

Multiply $2$ by $3$ .

$\frac{3 - 2 \sqrt{6} + 2 \sqrt{2} (- \sqrt{3}) + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.4

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

Tap for more steps...

Step 1.3.1.4.1

Multiply $- 1$ by $2$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{2} \sqrt{3} + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.4.2

Combine using the product rule for radicals.

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{3 \cdot 2} + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.4.3

Multiply $3$ by $2$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 2 \sqrt{2} (2 \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.5

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

Tap for more steps...

Step 1.3.1.5.1

Multiply $2$ by $2$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 \sqrt{2} \sqrt{2} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 ({\sqrt{2}}^{1} \sqrt{2}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.5.4

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

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 {\sqrt{2}}^{1 + 1} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.5.5

Add $1$ and $1$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 {\sqrt{2}}^{2} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.6

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

Tap for more steps...

Step 1.3.1.6.1

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

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 {(2^{\frac{1}{2}})}^{2} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.6.2

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

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 \cdot 2^{\frac{1}{2} \cdot 2} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.6.3

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

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 \cdot 2^{\frac{2}{2}} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.1.6.4.1

Cancel the common factor.

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 \cdot 2^{\frac{2}{2}} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.6.4.2

Rewrite the expression.

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 \cdot 2^{1} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.6.5

Evaluate the exponent.

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 4 \cdot 2 - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.1.7

Multiply $4$ by $2$ .

$\frac{3 - 2 \sqrt{6} - 2 \sqrt{6} + 8 - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.2

Add $3$ and $8$ .

$\frac{11 - 2 \sqrt{6} - 2 \sqrt{6} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.3.3

Subtract $2 \sqrt{6}$ from $- 2 \sqrt{6}$ .

$\frac{11 - 4 \sqrt{6} - (- 1 - \sqrt{3}) + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.4

Apply the distributive property.

$\frac{11 - 4 \sqrt{6} - - 1 - - \sqrt{3} + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.5

Multiply $- 1$ by $- 1$ .

$\frac{11 - 4 \sqrt{6} + 1 - - \sqrt{3} + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.6

Multiply $- - \sqrt{3}$ .

Tap for more steps...

Step 1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{11 - 4 \sqrt{6} + 1 + 1 \sqrt{3} + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.6.2

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

$\frac{11 - 4 \sqrt{6} + 1 + \sqrt{3} + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.7

Add $11$ and $1$ .

$\frac{12 - 4 \sqrt{6} + \sqrt{3} + 4 \sqrt{6}}{1 - \sqrt{3}}$

Step 1.8

Add $- 4 \sqrt{6}$ and $4 \sqrt{6}$ .

$\frac{12 + 0 + \sqrt{3}}{1 - \sqrt{3}}$

Step 1.9

Add $12$ and $0$ .

$\frac{12 + \sqrt{3}}{1 - \sqrt{3}}$

Step 2

Multiply $\frac{12 + \sqrt{3}}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

$\frac{12 + \sqrt{3}}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}}$

Step 3

Combine fractions.

Tap for more steps...

Step 3.1

Multiply $\frac{12 + \sqrt{3}}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

$\frac{(12 + \sqrt{3}) (1 + \sqrt{3})}{(1 - \sqrt{3}) (1 + \sqrt{3})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(12 + \sqrt{3}) (1 + \sqrt{3})}{1 + \sqrt{3} - \sqrt{3} - {\sqrt{3}}^{2}}$

Step 3.3

Simplify.

$\frac{(12 + \sqrt{3}) (1 + \sqrt{3})}{- 2}$

Step 4

Expand $(12 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 4.1

Apply the distributive property.

$\frac{12 (1 + \sqrt{3}) + \sqrt{3} (1 + \sqrt{3})}{- 2}$

Step 4.2

Apply the distributive property.

$\frac{12 \cdot 1 + 12 \sqrt{3} + \sqrt{3} (1 + \sqrt{3})}{- 2}$

Step 4.3

Apply the distributive property.

$\frac{12 \cdot 1 + 12 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}}{- 2}$

Step 5

Simplify and combine like terms.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

Multiply $12$ by $1$ .

$\frac{12 + 12 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}}{- 2}$

Step 5.1.2

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

$\frac{12 + 12 \sqrt{3} + \sqrt{3} + \sqrt{3} \sqrt{3}}{- 2}$

Step 5.1.3

Combine using the product rule for radicals.

$\frac{12 + 12 \sqrt{3} + \sqrt{3} + \sqrt{3 \cdot 3}}{- 2}$

Step 5.1.4

Multiply $3$ by $3$ .

$\frac{12 + 12 \sqrt{3} + \sqrt{3} + \sqrt{9}}{- 2}$

Step 5.1.5

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

$\frac{12 + 12 \sqrt{3} + \sqrt{3} + \sqrt{3^{2}}}{- 2}$

Step 5.1.6

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

$\frac{12 + 12 \sqrt{3} + \sqrt{3} + 3}{- 2}$

Step 5.2

Add $12$ and $3$ .

$\frac{15 + 12 \sqrt{3} + \sqrt{3}}{- 2}$

Step 5.3

Add $12 \sqrt{3}$ and $\sqrt{3}$ .

$\frac{15 + 13 \sqrt{3}}{- 2}$

Step 6

Move the negative in front of the fraction.

$- \frac{15 + 13 \sqrt{3}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{15 + 13 \sqrt{3}}{2}$

Decimal Form:

$- 18.75833024 \dots$