Pre-Algebra Examples

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

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

Step 1

Multiply

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

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

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

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

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

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

Step 2

Combine fractions.

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Step 2.1

Multiply

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

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

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

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

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

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

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}

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

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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