Algebra Examples

$\frac{6 - \sqrt{2}}{4 - 3 \sqrt{2}}$

Step 1

Multiply $\frac{6 - \sqrt{2}}{4 - 3 \sqrt{2}}$ by $\frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{6 - \sqrt{2}}{4 - 3 \sqrt{2}}$ by $\frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{2}) (4 + 3 \sqrt{2})}{16 + 12 \sqrt{2} - 12 \sqrt{2} - 9 {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

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

Step 3

Expand $(6 - \sqrt{2}) (4 + 3 \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $6$ by $4$ .

$\frac{24 + 6 (3 \sqrt{2}) - \sqrt{2} \cdot 4 - \sqrt{2} (3 \sqrt{2})}{- 2}$

Step 4.1.2

Multiply $3$ by $6$ .

$\frac{24 + 18 \sqrt{2} - \sqrt{2} \cdot 4 - \sqrt{2} (3 \sqrt{2})}{- 2}$

Step 4.1.3

Multiply $4$ by $- 1$ .

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - \sqrt{2} (3 \sqrt{2})}{- 2}$

Step 4.1.4

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

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Step 4.1.4.1

Multiply $3$ by $- 1$ .

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 \sqrt{2} \sqrt{2}}{- 2}$

Step 4.1.4.2

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

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 ({\sqrt{2}}^{1} \sqrt{2})}{- 2}$

Step 4.1.4.3

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

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 2}$

Step 4.1.4.4

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

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 {\sqrt{2}}^{1 + 1}}{- 2}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 {\sqrt{2}}^{2}}{- 2}$

Step 4.1.5

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

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Step 4.1.5.1

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

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 {(2^{\frac{1}{2}})}^{2}}{- 2}$

Step 4.1.5.2

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

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 \cdot 2^{\frac{1}{2} \cdot 2}}{- 2}$

Step 4.1.5.3

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

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 \cdot 2^{\frac{2}{2}}}{- 2}$

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 \cdot 2^{\frac{2}{2}}}{- 2}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 \cdot 2^{1}}{- 2}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 3 \cdot 2}{- 2}$

Step 4.1.6

Multiply $- 3$ by $2$ .

$\frac{24 + 18 \sqrt{2} - 4 \sqrt{2} - 6}{- 2}$

Step 4.2

Subtract $6$ from $24$ .

$\frac{18 + 18 \sqrt{2} - 4 \sqrt{2}}{- 2}$

Step 4.3

Subtract $4 \sqrt{2}$ from $18 \sqrt{2}$ .

$\frac{18 + 14 \sqrt{2}}{- 2}$

Step 5

Cancel the common factor of $18 + 14 \sqrt{2}$ and $- 2$ .

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Step 5.1

Factor $2$ out of $18$ .

$\frac{2 (9) + 14 \sqrt{2}}{- 2}$

Step 5.2

Factor $2$ out of $14 \sqrt{2}$ .

$\frac{2 (9) + 2 (7 \sqrt{2})}{- 2}$

Step 5.3

Factor $2$ out of $2 (9) + 2 (7 \sqrt{2})$ .

$\frac{2 (9 + 7 \sqrt{2})}{- 2}$

Step 5.4

Move the negative one from the denominator of $\frac{9 + 7 \sqrt{2}}{- 1}$ .

$- 1 \cdot (9 + 7 \sqrt{2})$

Step 6

Rewrite $- 1 \cdot (9 + 7 \sqrt{2})$ as $- (9 + 7 \sqrt{2})$ .

$- (9 + 7 \sqrt{2})$

Step 7

Apply the distributive property.

$- 1 \cdot 9 - (7 \sqrt{2})$

Step 8

Multiply.

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Step 8.1

Multiply $- 1$ by $9$ .

$- 9 - (7 \sqrt{2})$

Step 8.2

Multiply $7$ by $- 1$ .

$- 9 - 7 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 9 - 7 \sqrt{2}$

Decimal Form:

$- 18.89949493 \dots$