Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

Multiply $2$ by $3$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

$\frac{2 \sqrt{2 x} + \sqrt{6 x^{1 + 1}}}{2 x}$

Step 4.6

Add $1$ and $1$ .

$\frac{2 \sqrt{2 x} + \sqrt{6 x^{2}}}{2 x}$

Step 5

Simplify each term.

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Step 5.1

Reorder $6$ and $x^{2}$ .

$\frac{2 \sqrt{2 x} + \sqrt{x^{2} \cdot 6}}{2 x}$

Step 5.2

Pull terms out from under the radical.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 6.3

Apply the product rule to $2 x$ .

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

Step 6.4

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

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Step 6.4.1

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

$\frac{2^{\frac{1}{2}} \cdot 2 x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}}{2 x}$

Step 6.4.2

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

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Step 6.4.2.1

Raise $2$ to the power of $1$ .

$\frac{2^{\frac{1}{2}} \cdot 2^{1} x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}}{2 x}$

Step 6.4.2.2

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

$\frac{2^{\frac{1}{2} + 1} x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}}{2 x}$

Step 6.4.3

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

$\frac{2^{\frac{1}{2} + \frac{2}{2}} x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}}{2 x}$

Step 6.4.4

Combine the numerators over the common denominator.

$\frac{2^{\frac{1 + 2}{2}} x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}}{2 x}$

Step 6.4.5

Add $1$ and $2$ .

$\frac{2^{\frac{3}{2}} x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}}{2 x}$

Step 6.5

Factor $x^{\frac{1}{2}}$ out of $2^{\frac{3}{2}} x^{\frac{1}{2}} + x \cdot 6^{\frac{1}{2}}$ .

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Step 6.5.1

Reorder $x$ and $6^{\frac{1}{2}}$ .

$\frac{2^{\frac{3}{2}} x^{\frac{1}{2}} + 6^{\frac{1}{2}} \cdot x}{2 x}$

Step 6.5.2

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

$\frac{x^{\frac{1}{2}} \cdot 2^{\frac{3}{2}} + 6^{\frac{1}{2}} \cdot x}{2 x}$

Step 6.5.3

Factor $x^{\frac{1}{2}}$ out of $6^{\frac{1}{2}} \cdot x$ .

$\frac{x^{\frac{1}{2}} \cdot 2^{\frac{3}{2}} + x^{\frac{1}{2}} (6^{\frac{1}{2}} \cdot x^{\frac{1}{2}})}{2 x}$

Step 6.5.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 2^{\frac{3}{2}} + x^{\frac{1}{2}} (6^{\frac{1}{2}} \cdot x^{\frac{1}{2}})$ .

$\frac{x^{\frac{1}{2}} (2^{\frac{3}{2}} + 6^{\frac{1}{2}} \cdot x^{\frac{1}{2}})}{2 x}$

$\frac{x^{\frac{1}{2}} (2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x}$

Step 7

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

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x \cdot x^{- \frac{1}{2}}}$

Step 8

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

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Step 8.1

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

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 (x^{- \frac{1}{2}} x)}$

Step 8.2

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

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Step 8.2.1

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

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 (x^{- \frac{1}{2}} x^{1})}$

Step 8.2.2

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

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{- \frac{1}{2} + 1}}$

Step 8.3

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

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{- \frac{1}{2} + \frac{2}{2}}}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{\frac{- 1 + 2}{2}}}$

Step 8.5

Add $- 1$ and $2$ .

$\frac{2^{\frac{3}{2}} + 6^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$