Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} - \frac{1}{\sqrt{8}} + \sqrt{2}$

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Evaluate the exponent.

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

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

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Step 1.3.1.1

Factor $4$ out of $8$ .

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

Step 1.3.1.2

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

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

Step 1.3.2

Pull terms out from under the radical.

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

Step 1.4

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

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

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

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

Step 1.5.2

Move $\sqrt{2}$ .

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

Add $1$ and $1$ .

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

Step 1.5.7

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

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Step 1.5.7.1

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

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

Step 1.5.7.2

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

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

Step 1.5.7.3

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

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

Step 1.5.7.4

Cancel the common factor of $2$ .

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Step 1.5.7.4.1

Cancel the common factor.

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

Step 1.5.7.4.2

Rewrite the expression.

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

Step 1.5.7.5

Evaluate the exponent.

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

Step 1.6

Multiply $2$ by $2$ .

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write $\sqrt{2}$ as a fraction with denominator $1$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $2$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Move $2$ to the left of $\sqrt{2}$ .

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

Step 4.2

Move $4$ to the left of $\sqrt{2}$ .

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

Step 5

Simplify by adding terms.

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Step 5.1

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

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

Step 5.2

Add $\sqrt{2}$ and $4 \sqrt{2}$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.76776695 \dots$