Calculus Examples

$\frac{2 \sqrt{6^{3}}}{5} - \frac{2}{3 \sqrt{6^{3}}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Raise $6$ to the power of $3$ .

$\frac{2 \sqrt{216}}{5} - \frac{2}{3 \sqrt{6^{3}}}$

Step 1.1.2

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 1.1.2.1

Factor $36$ out of $216$ .

$\frac{2 \sqrt{36 (6)}}{5} - \frac{2}{3 \sqrt{6^{3}}}$

Step 1.1.2.2

Rewrite $36$ as $6^{2}$ .

$\frac{2 \sqrt{6^{2} \cdot 6}}{5} - \frac{2}{3 \sqrt{6^{3}}}$

Step 1.1.3

Pull terms out from under the radical.

$\frac{2 \cdot 6 \sqrt{6}}{5} - \frac{2}{3 \sqrt{6^{3}}}$

Step 1.1.4

Multiply $2$ by $6$ .

$\frac{12 \sqrt{6}}{5} - \frac{2}{3 \sqrt{6^{3}}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Raise $6$ to the power of $3$ .

$\frac{12 \sqrt{6}}{5} - \frac{2}{3 \sqrt{216}}$

Step 1.2.2

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 1.2.2.1

Factor $36$ out of $216$ .

$\frac{12 \sqrt{6}}{5} - \frac{2}{3 \sqrt{36 (6)}}$

Step 1.2.2.2

Rewrite $36$ as $6^{2}$ .

$\frac{12 \sqrt{6}}{5} - \frac{2}{3 \sqrt{6^{2} \cdot 6}}$

Step 1.2.3

Pull terms out from under the radical.

$\frac{12 \sqrt{6}}{5} - \frac{2}{3 \cdot 6 \sqrt{6}}$

Step 1.2.4

Multiply $3$ by $6$ .

$\frac{12 \sqrt{6}}{5} - \frac{2}{18 \sqrt{6}}$

Step 1.3

Cancel the common factor of $2$ and $18$ .

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Step 1.3.1

Factor $2$ out of $2$ .

$\frac{12 \sqrt{6}}{5} - \frac{2 (1)}{18 \sqrt{6}}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $2$ out of $18 \sqrt{6}$ .

$\frac{12 \sqrt{6}}{5} - \frac{2 (1)}{2 (9 \sqrt{6})}$

Step 1.3.2.2

Cancel the common factor.

$\frac{12 \sqrt{6}}{5} - \frac{2 \cdot 1}{2 (9 \sqrt{6})}$

Step 1.3.2.3

Rewrite the expression.

$\frac{12 \sqrt{6}}{5} - \frac{1}{9 \sqrt{6}}$

Step 1.4

Multiply $\frac{1}{9 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{12 \sqrt{6}}{5} - (\frac{1}{9 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{1}{9 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 \sqrt{6} \sqrt{6}}$

Step 1.5.2

Move $\sqrt{6}$ .

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 (\sqrt{6} \sqrt{6})}$

Step 1.5.3

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

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 ({\sqrt{6}}^{1} \sqrt{6})}$

Step 1.5.4

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

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 1.5.5

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

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 {\sqrt{6}}^{1 + 1}}$

Step 1.5.6

Add $1$ and $1$ .

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 {\sqrt{6}}^{2}}$

Step 1.5.7

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

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Step 1.5.7.1

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

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 {(6^{\frac{1}{2}})}^{2}}$

Step 1.5.7.2

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

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 1.5.7.3

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

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 \cdot 6^{\frac{2}{2}}}$

Step 1.5.7.4

Cancel the common factor of $2$ .

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Step 1.5.7.4.1

Cancel the common factor.

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 \cdot 6^{\frac{2}{2}}}$

Step 1.5.7.4.2

Rewrite the expression.

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 \cdot 6^{1}}$

Step 1.5.7.5

Evaluate the exponent.

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{9 \cdot 6}$

Step 1.6

Multiply $9$ by $6$ .

$\frac{12 \sqrt{6}}{5} - \frac{\sqrt{6}}{54}$

Step 2

To write $\frac{12 \sqrt{6}}{5}$ as a fraction with a common denominator, multiply by $\frac{54}{54}$ .

$\frac{12 \sqrt{6}}{5} \cdot \frac{54}{54} - \frac{\sqrt{6}}{54}$

Step 3

To write $- \frac{\sqrt{6}}{54}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{12 \sqrt{6}}{5} \cdot \frac{54}{54} - \frac{\sqrt{6}}{54} \cdot \frac{5}{5}$

Step 4

Write each expression with a common denominator of $270$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{12 \sqrt{6}}{5}$ by $\frac{54}{54}$ .

$\frac{12 \sqrt{6} \cdot 54}{5 \cdot 54} - \frac{\sqrt{6}}{54} \cdot \frac{5}{5}$

Step 4.2

Multiply $5$ by $54$ .

$\frac{12 \sqrt{6} \cdot 54}{270} - \frac{\sqrt{6}}{54} \cdot \frac{5}{5}$

Step 4.3

Multiply $\frac{\sqrt{6}}{54}$ by $\frac{5}{5}$ .

$\frac{12 \sqrt{6} \cdot 54}{270} - \frac{\sqrt{6} \cdot 5}{54 \cdot 5}$

Step 4.4

Multiply $54$ by $5$ .

$\frac{12 \sqrt{6} \cdot 54}{270} - \frac{\sqrt{6} \cdot 5}{270}$

Step 5

Combine the numerators over the common denominator.

$\frac{12 \sqrt{6} \cdot 54 - \sqrt{6} \cdot 5}{270}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $54$ by $12$ .

$\frac{648 \sqrt{6} - \sqrt{6} \cdot 5}{270}$

Step 6.2

Multiply $5$ by $- 1$ .

$\frac{648 \sqrt{6} - 5 \sqrt{6}}{270}$

Step 6.3

Subtract $5 \sqrt{6}$ from $648 \sqrt{6}$ .

$\frac{643 \sqrt{6}}{270}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{643 \sqrt{6}}{270}$

Decimal Form:

$5.83341446 \dots$