Simplify (8+ square root of 6)/( square root of 2)

$\frac{8 + \sqrt{6}}{\sqrt{2}}$

Step 1

Multiply $\frac{8 + \sqrt{6}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{8 + \sqrt{6}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{8 + \sqrt{6}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{(8 + \sqrt{6}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$\frac{(8 + \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

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

$\frac{(8 + \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

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

$\frac{(8 + \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(8 + \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

$\frac{(8 + \sqrt{6}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$\frac{(8 + \sqrt{6}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$\frac{(8 + \sqrt{6}) \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(8 + \sqrt{6}) \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(8 + \sqrt{6}) \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\frac{(8 + \sqrt{6}) \sqrt{2}}{2}$

Apply the distributive property.

$\frac{8 \sqrt{2} + \sqrt{6} \sqrt{2}}{2}$

Combine using the product rule for radicals.

$\frac{8 \sqrt{2} + \sqrt{6 \cdot 2}}{2}$

Step 3

Simplify each term.

Multiply $6$ by $2$ .

$\frac{8 \sqrt{2} + \sqrt{12}}{2}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{8 \sqrt{2} + \sqrt{4 (3)}}{2}$

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

$\frac{8 \sqrt{2} + \sqrt{2^{2} \cdot 3}}{2}$

Pull terms out from under the radical.

$\frac{8 \sqrt{2} + 2 \sqrt{3}}{2}$

Step 4

Cancel the common factor of $8 \sqrt{2} + 2 \sqrt{3}$ and $2$ .

Factor $2$ out of $8 \sqrt{2}$ .

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

Factor $2$ out of $2 \sqrt{3}$ .

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

Factor $2$ out of $2 (4 \sqrt{2}) + 2 (\sqrt{3})$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (4 \sqrt{2} + \sqrt{3})}{2 (1)}$

Cancel the common factor.

$\frac{2 (4 \sqrt{2} + \sqrt{3})}{2 \cdot 1}$

Rewrite the expression.

$\frac{4 \sqrt{2} + \sqrt{3}}{1}$

Divide $4 \sqrt{2} + \sqrt{3}$ by $1$ .

$4 \sqrt{2} + \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{2} + \sqrt{3}$

Decimal Form:

$7.38890505 \dots$

Simplify (8+ square root of 6)/( square root of 2)

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