Algebra Examples

$\frac{\sqrt{2} - \sqrt{8}}{2 \sqrt{2} - 1}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

$\frac{\sqrt{2} - \sqrt{2^{2} \cdot 2}}{2 \sqrt{2} - 1}$

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 1.4

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

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

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

Step 5

Expand the denominator using the FOIL method.

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

Step 6

Simplify.

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

Step 7

Apply the distributive property.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.1.4

Cancel the common factor of $2$ .

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Step 10.1.4.1

Cancel the common factor.

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

Step 10.1.4.2

Rewrite the expression.

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

Step 10.1.5

Evaluate the exponent.

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

Step 10.2

Multiply $2$ by $2$ .

$- \frac{4 + \sqrt{2}}{7}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 + \sqrt{2}}{7}$

Decimal Form:

$- 0.77345908 \dots$