Pre-Algebra Examples

$(6 - \sqrt{3}) \div (3 + \sqrt{3})$

Step 1

Rewrite the division as a fraction.

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

Step 2

Multiply $\frac{6 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{6 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}$

Step 3

Multiply $\frac{6 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

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

Step 4

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}$

Step 5

Simplify.

$\frac{(6 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 6

Expand $(6 - \sqrt{3}) (3 - \sqrt{3})$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\frac{6 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $6$ by $3$ .

$\frac{18 + 6 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 7.1.2

Multiply $- 1$ by $6$ .

$\frac{18 - 6 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 7.1.3

Multiply $3$ by $- 1$ .

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

Step 7.1.4

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

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Step 7.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

Step 7.1.4.2

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}$

Step 7.1.4.3

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

Step 7.1.4.4

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

Step 7.1.4.5

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Step 7.1.4.6

Add $1$ and $1$ .

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}$

Step 7.1.5

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

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Step 7.1.5.1

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}$

Step 7.1.5.2

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}$

Step 7.1.5.3

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

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Step 7.1.5.4

Cancel the common factor of $2$ .

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Step 7.1.5.4.1

Cancel the common factor.

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Step 7.1.5.4.2

Rewrite the expression.

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Step 7.1.5.5

Evaluate the exponent.

$\frac{18 - 6 \sqrt{3} - 3 \sqrt{3} + 3}{6}$

Step 7.2

Add $18$ and $3$ .

$\frac{21 - 6 \sqrt{3} - 3 \sqrt{3}}{6}$

Step 7.3

Subtract $3 \sqrt{3}$ from $- 6 \sqrt{3}$ .

$\frac{21 - 9 \sqrt{3}}{6}$

Step 8

Cancel the common factor of $21 - 9 \sqrt{3}$ and $6$ .

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Step 8.1

Factor $3$ out of $21$ .

$\frac{3 (7) - 9 \sqrt{3}}{6}$

Step 8.2

Factor $3$ out of $- 9 \sqrt{3}$ .

$\frac{3 (7) + 3 (- 3 \sqrt{3})}{6}$

Step 8.3

Factor $3$ out of $3 (7) + 3 (- 3 \sqrt{3})$ .

$\frac{3 (7 - 3 \sqrt{3})}{6}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $3$ out of $6$ .

$\frac{3 (7 - 3 \sqrt{3})}{3 \cdot 2}$

Step 8.4.2

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{3})}{3 \cdot 2}$

Step 8.4.3

Rewrite the expression.

$\frac{7 - 3 \sqrt{3}}{2}$

Step 9

Split the fraction $\frac{7 - 3 \sqrt{3}}{2}$ into two fractions.

$\frac{7}{2} + \frac{- 3 \sqrt{3}}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{7}{2} - \frac{3 \sqrt{3}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{2} - \frac{3 \sqrt{3}}{2}$

Decimal Form:

$0.90192378 \dots$