Algebra Examples

$\frac{3 + \sqrt{18}}{1 + \sqrt{8}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 1.1.1

Factor $9$ out of $18$ .

$\frac{3 + \sqrt{9 (2)}}{1 + \sqrt{8}}$

Step 1.1.2

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

$\frac{3 + \sqrt{3^{2} \cdot 2}}{1 + \sqrt{8}}$

Step 1.2

Pull terms out from under the radical.

$\frac{3 + 3 \sqrt{2}}{1 + \sqrt{8}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.1

Factor $4$ out of $8$ .

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

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

$\frac{3 + 3 \sqrt{2}}{1 + 2 \sqrt{2}}$

Step 3

Multiply $\frac{3 + 3 \sqrt{2}}{1 + 2 \sqrt{2}}$ by $\frac{1 - 2 \sqrt{2}}{1 - 2 \sqrt{2}}$ .

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

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{3 + 3 \sqrt{2}}{1 + 2 \sqrt{2}}$ by $\frac{1 - 2 \sqrt{2}}{1 - 2 \sqrt{2}}$ .

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

Step 4.2

Expand the denominator using the FOIL method.

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

Step 4.3

Simplify.

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

Step 5

Expand $(3 + 3 \sqrt{2}) (1 - 2 \sqrt{2})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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 $1$ .

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

Step 6.1.2

Multiply $- 2$ by $3$ .

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} \cdot 1 + 3 \sqrt{2} (- 2 \sqrt{2})}{- 7}$

Step 6.1.3

Multiply $3$ by $1$ .

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} + 3 \sqrt{2} (- 2 \sqrt{2})}{- 7}$

Step 6.1.4

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

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Step 6.1.4.1

Multiply $- 2$ by $3$ .

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 \sqrt{2} \sqrt{2}}{- 7}$

Step 6.1.4.2

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

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 ({\sqrt{2}}^{1} \sqrt{2})}{- 7}$

Step 6.1.4.3

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

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 7}$

Step 6.1.4.4

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

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 {\sqrt{2}}^{1 + 1}}{- 7}$

Step 6.1.4.5

Add $1$ and $1$ .

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 {\sqrt{2}}^{2}}{- 7}$

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 $2^{\frac{1}{2}}$ .

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 {(2^{\frac{1}{2}})}^{2}}{- 7}$

Step 6.1.5.2

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

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 \cdot 2^{\frac{1}{2} \cdot 2}}{- 7}$

Step 6.1.5.3

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

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 \cdot 2^{\frac{2}{2}}}{- 7}$

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.

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 \cdot 2^{\frac{2}{2}}}{- 7}$

Step 6.1.5.4.2

Rewrite the expression.

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 \cdot 2^{1}}{- 7}$

Step 6.1.5.5

Evaluate the exponent.

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 6 \cdot 2}{- 7}$

Step 6.1.6

Multiply $- 6$ by $2$ .

$\frac{3 - 6 \sqrt{2} + 3 \sqrt{2} - 12}{- 7}$

Step 6.2

Subtract $12$ from $3$ .

$\frac{- 9 - 6 \sqrt{2} + 3 \sqrt{2}}{- 7}$

Step 6.3

Add $- 6 \sqrt{2}$ and $3 \sqrt{2}$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Rewrite $- 9$ as $- 1 (9)$ .

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

Step 9

Factor $- 1$ out of $- 3 \sqrt{2}$ .

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

Step 10

Factor $- 1$ out of $- 1 (9) - (3 \sqrt{2})$ .

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

Step 11

Simplify the expression.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

Multiply $- 1$ by $- 1$ .

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

Step 11.3

Multiply $\frac{9 + 3 \sqrt{2}}{7}$ by $1$ .

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.89180581 \dots$