Algebra Examples

$\frac{1 - \sqrt{3 x}}{\sqrt{6 x}}$

Step 1

Multiply $\frac{1 - \sqrt{3 x}}{\sqrt{6 x}}$ by $\frac{\sqrt{6 x}}{\sqrt{6 x}}$ .

$\frac{1 - \sqrt{3 x}}{\sqrt{6 x}} \cdot \frac{\sqrt{6 x}}{\sqrt{6 x}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1 - \sqrt{3 x}}{\sqrt{6 x}}$ by $\frac{\sqrt{6 x}}{\sqrt{6 x}}$ .

$\frac{(1 - \sqrt{3 x}) \sqrt{6 x}}{\sqrt{6 x} \sqrt{6 x}}$

Step 2.2

Raise $\sqrt{6 x}$ to the power of $1$ .

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

Step 2.3

Raise $\sqrt{6 x}$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

Rewrite ${\sqrt{6 x}}^{2}$ as $6 x$ .

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

$\frac{(1 - \sqrt{3 x}) \sqrt{6 x}}{6 x}$

Step 3

Apply the distributive property.

$\frac{1 \sqrt{6 x} - \sqrt{3 x} \sqrt{6 x}}{6 x}$

Step 4

Multiply $\sqrt{6 x}$ by $1$ .

$\frac{\sqrt{6 x} - \sqrt{3 x} \sqrt{6 x}}{6 x}$

Step 5

Multiply $- \sqrt{3 x} \sqrt{6 x}$ .

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Step 5.1

Combine using the product rule for radicals.

$\frac{\sqrt{6 x} - \sqrt{6 x (3 x)}}{6 x}$

Step 5.2

Multiply $3$ by $6$ .

$\frac{\sqrt{6 x} - \sqrt{18 x \cdot x}}{6 x}$

Step 5.3

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

$\frac{\sqrt{6 x} - \sqrt{18 (x^{1} x)}}{6 x}$

Step 5.4

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

$\frac{\sqrt{6 x} - \sqrt{18 (x^{1} x^{1})}}{6 x}$

Step 5.5

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

$\frac{\sqrt{6 x} - \sqrt{18 x^{1 + 1}}}{6 x}$

Step 5.6

Add $1$ and $1$ .

$\frac{\sqrt{6 x} - \sqrt{18 x^{2}}}{6 x}$

Step 6

Simplify each term.

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Step 6.1

Rewrite $18 x^{2}$ as ${(3 x)}^{2} \cdot 2$ .

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Step 6.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{6 x} - \sqrt{9 (2) x^{2}}}{6 x}$

Step 6.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 6.1.3

Move $2$ .

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

Step 6.1.4

Rewrite $3^{2} x^{2}$ as ${(3 x)}^{2}$ .

$\frac{\sqrt{6 x} - \sqrt{{(3 x)}^{2} \cdot 2}}{6 x}$

Step 6.2

Pull terms out from under the radical.

$\frac{\sqrt{6 x} - (3 x \sqrt{2})}{6 x}$

Step 6.3

Multiply $3$ by $- 1$ .

$\frac{\sqrt{6 x} - 3 x \sqrt{2}}{6 x}$

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Apply the product rule to $6 x$ .

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

Step 7.4

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

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Step 7.4.1

Move $x$ .

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

Step 7.4.2

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

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

Step 7.4.3

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

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

Step 7.4.4

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.3

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

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

Step 9.4

Combine the numerators over the common denominator.

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

Step 9.5

Add $- 1$ and $2$ .

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