Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Raise $3 - \sqrt{2}$ to the power of $1$ .

$\frac{{(3 - \sqrt{2})}^{1} (3 - \sqrt{2})}{7}$

Step 3.2

Raise $3 - \sqrt{2}$ to the power of $1$ .

$\frac{{(3 - \sqrt{2})}^{1} {(3 - \sqrt{2})}^{1}}{7}$

Step 3.3

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

$\frac{{(3 - \sqrt{2})}^{1 + 1}}{7}$

Step 3.4

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 (- \sqrt{2}) - \sqrt{2} \cdot 3 - \sqrt{2} (- \sqrt{2})}{7}$

Step 6.1.2

Multiply $- 1$ by $3$ .

$\frac{9 - 3 \sqrt{2} - \sqrt{2} \cdot 3 - \sqrt{2} (- \sqrt{2})}{7}$

Step 6.1.3

Multiply $3$ by $- 1$ .

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} - \sqrt{2} (- \sqrt{2})}{7}$

Step 6.1.4

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

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Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + 1 \sqrt{2} \sqrt{2}}{7}$

Step 6.1.4.2

Multiply $\sqrt{2}$ by $1$ .

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + \sqrt{2} \sqrt{2}}{7}$

Step 6.1.4.3

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

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + {\sqrt{2}}^{1} \sqrt{2}}{7}$

Step 6.1.4.4

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

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{7}$

Step 6.1.4.5

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

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + {\sqrt{2}}^{1 + 1}}{7}$

Step 6.1.4.6

Add $1$ and $1$ .

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + {\sqrt{2}}^{2}}{7}$

Step 6.1.5

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

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Step 6.1.5.1

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

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + {(2^{\frac{1}{2}})}^{2}}{7}$

Step 6.1.5.2

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

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + 2^{\frac{1}{2} \cdot 2}}{7}$

Step 6.1.5.3

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

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + 2^{\frac{2}{2}}}{7}$

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + 2^{\frac{2}{2}}}{7}$

Step 6.1.5.4.2

Rewrite the expression.

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + 2^{1}}{7}$

Step 6.1.5.5

Evaluate the exponent.

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2} + 2}{7}$

Step 6.2

Add $9$ and $2$ .

$\frac{11 - 3 \sqrt{2} - 3 \sqrt{2}}{7}$

Step 6.3

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

$\frac{11 - 6 \sqrt{2}}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{11 - 6 \sqrt{2}}{7}$

Decimal Form:

$0.35924551 \dots$