Simplify (5- square root of 8)/( square root of 2)

$\frac{5 - \sqrt{8}}{\sqrt{2}}$

Step 1

Simplify the numerator.

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

Factor $4$ out of $8$ .

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

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

$\frac{5 - \sqrt{2^{2} \cdot 2}}{\sqrt{2}}$

Pull terms out from under the radical.

$\frac{5 - (2 \sqrt{2})}{\sqrt{2}}$

Multiply $2$ by $- 1$ .

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

Step 2

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

$\frac{5 - 2 \sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\frac{(5 - 2 \sqrt{2}) \sqrt{2}}{2}$

Step 4

Apply the distributive property.

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

Step 5

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

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

$\frac{5 \sqrt{2} - 2 ({\sqrt{2}}^{1} \sqrt{2})}{2}$

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

$\frac{5 \sqrt{2} - 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2}$

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

$\frac{5 \sqrt{2} - 2 {\sqrt{2}}^{1 + 1}}{2}$

Add $1$ and $1$ .

$\frac{5 \sqrt{2} - 2 {\sqrt{2}}^{2}}{2}$

Step 6

Simplify each term.

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

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

$\frac{5 \sqrt{2} - 2 {(2^{\frac{1}{2}})}^{2}}{2}$

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

$\frac{5 \sqrt{2} - 2 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

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

$\frac{5 \sqrt{2} - 2 \cdot 2^{\frac{2}{2}}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 \sqrt{2} - 2 \cdot 2^{\frac{2}{2}}}{2}$

Rewrite the expression.

$\frac{5 \sqrt{2} - 2 \cdot 2^{1}}{2}$

Evaluate the exponent.

$\frac{5 \sqrt{2} - 2 \cdot 2}{2}$

Multiply $- 2$ by $2$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.53553390 \dots$

Simplify (5- square root of 8)/( square root of 2)

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