Simplify ( square root of 3+3 square root of 5)/(2 square root of 8)

$\frac{\sqrt{3} + 3 \sqrt{5}}{2 \sqrt{8}}$

Step 1

Simplify the denominator.

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

Factor $4$ out of $8$ .

$\frac{\sqrt{3} + 3 \sqrt{5}}{2 \sqrt{4 (2)}}$

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

$\frac{\sqrt{3} + 3 \sqrt{5}}{2 \sqrt{2^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{\sqrt{3} + 3 \sqrt{5}}{2 \cdot 2 \sqrt{2}}$

Multiply $2$ by $2$ .

$\frac{\sqrt{3} + 3 \sqrt{5}}{4 \sqrt{2}}$

Step 2

Multiply $\frac{\sqrt{3} + 3 \sqrt{5}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{3} + 3 \sqrt{5}}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3} + 3 \sqrt{5}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

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

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})}$

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

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 {\sqrt{2}}^{2}}$

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

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

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 \cdot 2^{1}}$

Evaluate the exponent.

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{4 \cdot 2}$

Step 4

Multiply $4$ by $2$ .

$\frac{(\sqrt{3} + 3 \sqrt{5}) \sqrt{2}}{8}$

Step 5

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{2} + 3 \sqrt{5} \sqrt{2}}{8}$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 2} + 3 \sqrt{5} \sqrt{2}}{8}$

Step 7

Multiply $3 \sqrt{5} \sqrt{2}$ .

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 2} + 3 \sqrt{2 \cdot 5}}{8}$

Multiply $2$ by $5$ .

$\frac{\sqrt{3 \cdot 2} + 3 \sqrt{10}}{8}$

Step 8

Multiply $3$ by $2$ .

$\frac{\sqrt{6} + 3 \sqrt{10}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6} + 3 \sqrt{10}}{8}$

Decimal Form:

$1.49204034 \dots$

Simplify ( square root of 3+3 square root of 5)/(2 square root of 8)

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