Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

Move $\sqrt{2 x}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Simplify.

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

Step 4

Multiply $2$ by $2$ .

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

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

Step 7.5

Simplify.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

Apply the product rule to $2 x$ .

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.3

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

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

Step 10.4

Combine the numerators over the common denominator.

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

Step 10.5

Add $- 1$ and $2$ .

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

Step 11

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

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

Step 12

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

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

Step 13

Cancel the common factors.

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Step 13.1

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

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

Step 13.2

Cancel the common factor.

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

Step 13.3

Rewrite the expression.

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

Step 14

Cancel the common factor.

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

Step 15

Rewrite the expression.

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