Divide ( square root of 2x+3)/( square root of 8x)

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

Simplify the denominator.

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{2 x}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $2$ by $2$ .

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

Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{{\sqrt{2 x}}^{2} + 3 \sqrt{2 x}}{4 x}$

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $2 x$ .

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

Factor $x^{\frac{1}{2}}$ out of $2 x + 3 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

Factor $x^{\frac{1}{2}}$ out of $2 x$ .

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

Factor $x^{\frac{1}{2}}$ out of $3 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (2 x^{\frac{1}{2}}) + x^{\frac{1}{2}} (3 \cdot 2^{\frac{1}{2}})$ .

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

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 x \cdot x^{- \frac{1}{2}}}$

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Move $x^{- \frac{1}{2}}$ .

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 (x^{- \frac{1}{2}} x)}$

Multiply $x^{- \frac{1}{2}}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 (x^{- \frac{1}{2}} x^{1})}$

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

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 x^{- \frac{1}{2} + 1}}$

Write $1$ as a fraction with a common denominator.

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 x^{- \frac{1}{2} + \frac{2}{2}}}$

Combine the numerators over the common denominator.

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{- 1 + 2}{2}}}$

Add $- 1$ and $2$ .

$\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{1}{2}}}$

Split the fraction $\frac{2 x^{\frac{1}{2}} + 3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{1}{2}}}$ into two fractions.

$\frac{2 x^{\frac{1}{2}}}{4 x^{\frac{1}{2}}} + \frac{3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{1}{2}}}$

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Cancel the common factors.

Factor $2$ out of $4 x^{\frac{1}{2}}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{1}{2}}}$

Cancel the common factor.

$\frac{x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{1}{2} + \frac{3 \cdot 2^{\frac{1}{2}}}{4 x^{\frac{1}{2}}}$

Divide ( square root of 2x+3)/( square root of 8x)

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