Algebra Examples

$3 \sqrt{24 x} - 6 \sqrt{\frac{x}{6}} - \sqrt{54 x}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $24 x$ as $2^{2} \cdot (6 x)$ .

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Step 1.1.1

Factor $4$ out of $24$ .

$3 \sqrt{4 (6) x} - 6 \sqrt{\frac{x}{6}} - \sqrt{54 x}$

Step 1.1.2

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

$3 \sqrt{2^{2} \cdot 6 x} - 6 \sqrt{\frac{x}{6}} - \sqrt{54 x}$

Step 1.1.3

Add parentheses.

$3 \sqrt{2^{2} \cdot (6 x)} - 6 \sqrt{\frac{x}{6}} - \sqrt{54 x}$

Step 1.2

Pull terms out from under the radical.

$3 (2 \sqrt{6 x}) - 6 \sqrt{\frac{x}{6}} - \sqrt{54 x}$

Step 1.3

Multiply $2$ by $3$ .

$6 \sqrt{6 x} - 6 \sqrt{\frac{x}{6}} - \sqrt{54 x}$

Step 1.4

Rewrite $\sqrt{\frac{x}{6}}$ as $\frac{\sqrt{x}}{\sqrt{6}}$ .

$6 \sqrt{6 x} - 6 \frac{\sqrt{x}}{\sqrt{6}} - \sqrt{54 x}$

Step 1.5

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

$6 \sqrt{6 x} - 6 (\frac{\sqrt{x}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}) - \sqrt{54 x}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{\sqrt{6} \sqrt{6}} - \sqrt{54 x}$

Step 1.6.2

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}} - \sqrt{54 x}$

Step 1.6.3

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}} - \sqrt{54 x}$

Step 1.6.4

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{1 + 1}} - \sqrt{54 x}$

Step 1.6.5

Add $1$ and $1$ .

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{2}} - \sqrt{54 x}$

Step 1.6.6

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

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Step 1.6.6.1

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} - \sqrt{54 x}$

Step 1.6.6.2

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}} - \sqrt{54 x}$

Step 1.6.6.3

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

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{6^{\frac{2}{2}}} - \sqrt{54 x}$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{6^{\frac{2}{2}}} - \sqrt{54 x}$

Step 1.6.6.4.2

Rewrite the expression.

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{6^{1}} - \sqrt{54 x}$

Step 1.6.6.5

Evaluate the exponent.

$6 \sqrt{6 x} - 6 \frac{\sqrt{x} \sqrt{6}}{6} - \sqrt{54 x}$

Step 1.7

Combine using the product rule for radicals.

$6 \sqrt{6 x} - 6 \frac{\sqrt{x \cdot 6}}{6} - \sqrt{54 x}$

Step 1.8

Cancel the common factor of $6$ .

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Step 1.8.1

Factor $6$ out of $- 6$ .

$6 \sqrt{6 x} + 6 (- 1) \frac{\sqrt{x \cdot 6}}{6} - \sqrt{54 x}$

Step 1.8.2

Cancel the common factor.

$6 \sqrt{6 x} + 6 \cdot - 1 \frac{\sqrt{x \cdot 6}}{6} - \sqrt{54 x}$

Step 1.8.3

Rewrite the expression.

$6 \sqrt{6 x} - 1 \sqrt{x \cdot 6} - \sqrt{54 x}$

Step 1.9

Rewrite $- 1 \sqrt{x \cdot 6}$ as $- \sqrt{x \cdot 6}$ .

$6 \sqrt{6 x} - \sqrt{x \cdot 6} - \sqrt{54 x}$

Step 1.10

Rewrite $54 x$ as $3^{2} \cdot (6 x)$ .

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Step 1.10.1

Factor $9$ out of $54$ .

$6 \sqrt{6 x} - \sqrt{x \cdot 6} - \sqrt{9 (6) x}$

Step 1.10.2

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

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

Step 1.10.3

Add parentheses.

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

Step 1.11

Pull terms out from under the radical.

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

Step 1.12

Multiply $3$ by $- 1$ .

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

Step 2

Subtract $\sqrt{x \cdot 6}$ from $6 \sqrt{6 x}$ .

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Step 2.1

Reorder $x$ and $6$ .

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

Step 2.2

Subtract $\sqrt{6 \cdot x}$ from $6 \sqrt{6 x}$ .

$5 \sqrt{6 x} - 3 \sqrt{6 x}$

Step 3

Subtract $3 \sqrt{6 x}$ from $5 \sqrt{6 x}$ .

$2 \sqrt{6 x}$