Simplify (4- square root of 2)/(2 square root of 8)

$\frac{4 - \sqrt{2}}{2 \sqrt{8}}$

Step 1

Simplify the denominator.

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Move $\sqrt{2}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 4

Multiply $4$ by $2$ .

$\frac{(4 - \sqrt{2}) \sqrt{2}}{8}$

Step 5

Apply the distributive property.

$\frac{4 \sqrt{2} - \sqrt{2} \sqrt{2}}{8}$

Step 6

Multiply $- \sqrt{2} \sqrt{2}$ .

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

$\frac{4 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}{8}$

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

$\frac{4 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8}$

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

$\frac{4 \sqrt{2} - {\sqrt{2}}^{1 + 1}}{8}$

Add $1$ and $1$ .

$\frac{4 \sqrt{2} - {\sqrt{2}}^{2}}{8}$

Step 7

Simplify each term.

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

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

$\frac{4 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}{8}$

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

$\frac{4 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}{8}$

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

$\frac{4 \sqrt{2} - 2^{\frac{2}{2}}}{8}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 \sqrt{2} - 2^{\frac{2}{2}}}{8}$

Rewrite the expression.

$\frac{4 \sqrt{2} - 2^{1}}{8}$

Evaluate the exponent.

$\frac{4 \sqrt{2} - 1 \cdot 2}{8}$

Multiply $- 1$ by $2$ .

$\frac{4 \sqrt{2} - 2}{8}$

Step 8

Cancel the common factor of $4 \sqrt{2} - 2$ and $8$ .

Factor $2$ out of $4 \sqrt{2}$ .

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

Factor $2$ out of $- 2$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.45710678 \dots$

Simplify (4- square root of 2)/(2 square root of 8)

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