Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(1 + 2 \sqrt{2}) (2 \sqrt{2} + 3)}{4 {\sqrt{2}}^{2} + 6 \sqrt{2} - 6 \sqrt{2} - 9}$

Step 2.3

Simplify.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of $\frac{(1 + 2 \sqrt{2}) (2 \sqrt{2} + 3)}{- 1}$ .

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

Step 2.4.2

Rewrite $- 1 \cdot ((1 + 2 \sqrt{2}) (2 \sqrt{2} + 3))$ as $- ((1 + 2 \sqrt{2}) (2 \sqrt{2} + 3))$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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

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

Step 4.1.2

Multiply $3$ by $1$ .

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

Step 4.1.3

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

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Step 4.1.3.1

Multiply $2$ by $2$ .

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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

Step 4.1.3.5

Add $1$ and $1$ .

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

Step 4.1.4

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

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Step 4.1.4.1

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

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

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

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

Step 4.1.4.4.2

Rewrite the expression.

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

Step 4.1.4.5

Evaluate the exponent.

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

Step 4.1.5

Multiply $4$ by $2$ .

$- (2 \sqrt{2} + 3 + 8 + 2 \sqrt{2} \cdot 3)$

Step 4.1.6

Multiply $3$ by $2$ .

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

Step 4.2

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

$- (3 + 8 + 8 \sqrt{2})$

Step 4.3

Add $3$ and $8$ .

$- (11 + 8 \sqrt{2})$

Step 5

Apply the distributive property.

$- 1 \cdot 11 - (8 \sqrt{2})$

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $11$ .

$- 11 - (8 \sqrt{2})$

Step 6.2

Multiply $8$ by $- 1$ .

$- 11 - 8 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 11 - 8 \sqrt{2}$

Decimal Form:

$- 22.31370849 \dots$

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