Algebra Examples

$\frac{5 - 4 \sqrt{2}}{4 - 3 \sqrt{2}}$

Step 1

Multiply $\frac{5 - 4 \sqrt{2}}{4 - 3 \sqrt{2}}$ by $\frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$ .

$\frac{5 - 4 \sqrt{2}}{4 - 3 \sqrt{2}} \cdot \frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 - 4 \sqrt{2}}{4 - 3 \sqrt{2}}$ by $\frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$ .

$\frac{(5 - 4 \sqrt{2}) (4 + 3 \sqrt{2})}{(4 - 3 \sqrt{2}) (4 + 3 \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 - 4 \sqrt{2}) (4 + 3 \sqrt{2})}{16 + 12 \sqrt{2} - 12 \sqrt{2} - 9 {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 - 4 \sqrt{2}) (4 + 3 \sqrt{2})}{- 2}$

Step 3

Expand $(5 - 4 \sqrt{2}) (4 + 3 \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 (4 + 3 \sqrt{2}) - 4 \sqrt{2} (4 + 3 \sqrt{2})}{- 2}$

Step 3.2

Apply the distributive property.

$\frac{5 \cdot 4 + 5 (3 \sqrt{2}) - 4 \sqrt{2} (4 + 3 \sqrt{2})}{- 2}$

Step 3.3

Apply the distributive property.

$\frac{5 \cdot 4 + 5 (3 \sqrt{2}) - 4 \sqrt{2} \cdot 4 - 4 \sqrt{2} (3 \sqrt{2})}{- 2}$

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 $5$ by $4$ .

$\frac{20 + 5 (3 \sqrt{2}) - 4 \sqrt{2} \cdot 4 - 4 \sqrt{2} (3 \sqrt{2})}{- 2}$

Step 4.1.2

Multiply $3$ by $5$ .

$\frac{20 + 15 \sqrt{2} - 4 \sqrt{2} \cdot 4 - 4 \sqrt{2} (3 \sqrt{2})}{- 2}$

Step 4.1.3

Multiply $4$ by $- 4$ .

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 4 \sqrt{2} (3 \sqrt{2})}{- 2}$

Step 4.1.4

Multiply $- 4 \sqrt{2} (3 \sqrt{2})$ .

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Step 4.1.4.1

Multiply $3$ by $- 4$ .

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 \sqrt{2} \sqrt{2}}{- 2}$

Step 4.1.4.2

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

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 ({\sqrt{2}}^{1} \sqrt{2})}{- 2}$

Step 4.1.4.3

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

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 2}$

Step 4.1.4.4

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

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 {\sqrt{2}}^{1 + 1}}{- 2}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 {\sqrt{2}}^{2}}{- 2}$

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

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 {(2^{\frac{1}{2}})}^{2}}{- 2}$

Step 4.1.5.2

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

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 \cdot 2^{\frac{1}{2} \cdot 2}}{- 2}$

Step 4.1.5.3

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

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 \cdot 2^{\frac{2}{2}}}{- 2}$

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.

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 \cdot 2^{\frac{2}{2}}}{- 2}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 \cdot 2^{1}}{- 2}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 12 \cdot 2}{- 2}$

Step 4.1.6

Multiply $- 12$ by $2$ .

$\frac{20 + 15 \sqrt{2} - 16 \sqrt{2} - 24}{- 2}$

Step 4.2

Subtract $24$ from $20$ .

$\frac{- 4 + 15 \sqrt{2} - 16 \sqrt{2}}{- 2}$

Step 4.3

Subtract $16 \sqrt{2}$ from $15 \sqrt{2}$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 7

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

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

Step 8

Factor $- 1$ out of $- 1 (4) - (\sqrt{2})$ .

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

Step 9

Simplify the expression.

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Step 9.1

Move the negative in front of the fraction.

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

Step 9.2

Multiply $- 1$ by $- 1$ .

$1 \frac{4 + \sqrt{2}}{2}$

Step 9.3

Multiply $\frac{4 + \sqrt{2}}{2}$ by $1$ .

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.70710678 \dots$