Calculus Examples

$\int_{0}^{1} x^{3} \sqrt{3 + x^{2}} d x$

Step 1

Let $x = \sqrt{3} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sec^{2} (t) \sqrt{3} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sec^{2} (t) \sqrt{3}$ is positive.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{{\sqrt{3}}^{2} + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{\sqrt{3}}^{2} + {(\sqrt{3} \tan (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{{(3^{\frac{1}{2}})}^{2} + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.1.2

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3^{\frac{1}{2} \cdot 2} + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.1.3

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3^{\frac{2}{2}} + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.1.4.1

Cancel the common factor.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3^{\frac{2}{2}} + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.1.4.2

Rewrite the expression.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3^{1} + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.1.5

Evaluate the exponent.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + {(\sqrt{3} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.2

Apply the product rule to $\sqrt{3} \tan (t)$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + {\sqrt{3}}^{2} \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + {(3^{\frac{1}{2}})}^{2} \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.3.2

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + 3^{\frac{1}{2} \cdot 2} \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.3.3

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + 3^{\frac{2}{2}} \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.3.4.1

Cancel the common factor.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + 3^{\frac{2}{2}} \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.3.4.2

Rewrite the expression.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + 3^{1} \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.1.3.5

Evaluate the exponent.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 + 3 \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.2

Factor $3$ out of $3$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 (1) + 3 \tan^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.3

Factor $3$ out of $3 \tan^{2} (t)$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 (1) + 3 (\tan^{2} (t))} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.4

Factor $3$ out of $3 (1) + 3 (\tan^{2} (t))$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 (1 + \tan^{2} (t))} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.5

Rearrange terms.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 (\tan^{2} (t) + 1)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.6

Apply pythagorean identity.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{3 \sec^{2} (t)} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.7

Reorder $3$ and $\sec^{2} (t)$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sqrt{\sec^{2} (t) \cdot 3} (\sec^{2} (t) \sqrt{3}) d t$

Step 2.1.8

Pull terms out from under the radical.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} (\sec (t) \sqrt{3}) (\sec^{2} (t) \sqrt{3}) d t$

Step 2.2

Simplify.

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Step 2.2.1

Raise $\sec (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} (\sec^{2} (t) \sec^{1} (t)) \sqrt{3} \sqrt{3} d t$

Step 2.2.2

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} {\sec (t)}^{2 + 1} \sqrt{3} \sqrt{3} d t$

Step 2.2.3

Add $2$ and $1$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) \sqrt{3} \sqrt{3} d t$

Step 2.2.4

Raise $\sqrt{3}$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) ({\sqrt{3}}^{1} \sqrt{3}) d t$

Step 2.2.5

Raise $\sqrt{3}$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) d t$

Step 2.2.6

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) {\sqrt{3}}^{1 + 1} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) {\sqrt{3}}^{2} d t$

Step 2.2.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) {(3^{\frac{1}{2}})}^{2} d t$

Step 2.2.8.2

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) \cdot 3^{\frac{1}{2} \cdot 2} d t$

Step 2.2.8.3

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

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) \cdot 3^{\frac{2}{2}} d t$

Step 2.2.8.4

Cancel the common factor of $2$ .

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Step 2.2.8.4.1

Cancel the common factor.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) \cdot 3^{\frac{2}{2}} d t$

Step 2.2.8.4.2

Rewrite the expression.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) \cdot 3^{1} d t$

Step 2.2.8.5

Evaluate the exponent.

$\int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) \cdot 3 d t$

Step 2.2.9

Move $3$ to the left of ${(\sqrt{3} \tan (t))}^{3} \sec^{3} (t)$ .

$\int_{0}^{\frac{π}{6}} 3 {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) d t$

Step 3

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int_{0}^{\frac{π}{6}} {(\sqrt{3} \tan (t))}^{3} \sec^{3} (t) d t$

Step 4

Simplify the expression.

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Step 4.1

Apply the product rule to $\sqrt{3} \tan (t)$ .

$3 \int_{0}^{\frac{π}{6}} {\sqrt{3}}^{3} \tan^{3} (t) \sec^{3} (t) d t$

Step 4.2

Simplify.

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Step 4.2.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$3 \int_{0}^{\frac{π}{6}} \sqrt{3^{3}} \tan^{3} (t) \sec^{3} (t) d t$

Step 4.2.2

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

$3 \int_{0}^{\frac{π}{6}} \sqrt{27} \tan^{3} (t) \sec^{3} (t) d t$

Step 5

Since $\sqrt{27}$ is constant with respect to $t$ , move $\sqrt{27}$ out of the integral.

$3 (\sqrt{27} \int_{0}^{\frac{π}{6}} \tan^{3} (t) \sec^{3} (t) d t)$

Step 6

Factor out $\tan^{2} (t)$ .

$3 \sqrt{27} \int_{0}^{\frac{π}{6}} \tan^{2} (t) \tan (t) \sec^{3} (t) d t$

Step 7

Using the Pythagorean Identity, rewrite $\tan^{2} (t)$ as $- 1 + \sec^{2} (t)$ .

$3 \sqrt{27} \int_{0}^{\frac{π}{6}} (- 1 + \sec^{2} (t)) \tan (t) \sec^{3} (t) d t$

Step 8

Let $u = \sec (t)$ . Then $d u = \sec (t) \tan (t) d t$ , so $\frac{1}{\sec (t) \tan (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = \sec (t)$ . Find $\frac{d u}{d t}$ .

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Step 8.1.1

Differentiate $\sec (t)$ .

$\frac{d}{d t} [\sec (t)]$

Step 8.1.2

The derivative of $\sec (t)$ with respect to $t$ is $\sec (t) \tan (t)$ .

$\sec (t) \tan (t)$

Step 8.2

Substitute the lower limit in for $t$ in $u = \sec (t)$ .

$u_{lower} = \sec (0)$

Step 8.3

The exact value of $\sec (0)$ is $1$ .

$u_{lower} = 1$

Step 8.4

Substitute the upper limit in for $t$ in $u = \sec (t)$ .

$u_{upper} = \sec (\frac{π}{6})$

Step 8.5

Simplify.

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Step 8.5.1

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$u_{upper} = \frac{2}{\sqrt{3}}$

Step 8.5.2

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

$u_{upper} = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8.5.3

Combine and simplify the denominator.

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Step 8.5.3.1

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

$u_{upper} = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.5.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$u_{upper} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.5.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$u_{upper} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.5.3.4

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

$u_{upper} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.5.3.5

Add $1$ and $1$ .

$u_{upper} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.5.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$u_{upper} = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.5.3.6.2

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

$u_{upper} = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.5.3.6.3

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

$u_{upper} = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.5.3.6.4

Cancel the common factor of $2$ .

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Step 8.5.3.6.4.1

Cancel the common factor.

$u_{upper} = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.5.3.6.4.2

Rewrite the expression.

$u_{upper} = \frac{2 \sqrt{3}}{3^{1}}$

Step 8.5.3.6.5

Evaluate the exponent.

$u_{upper} = \frac{2 \sqrt{3}}{3}$

Step 8.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = \frac{2 \sqrt{3}}{3}$

Step 8.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$3 \sqrt{27} \int_{1}^{\frac{2 \sqrt{3}}{3}} (- 1 + u^{2}) u^{2} d u$

Step 9

Multiply $(- 1 + u^{2}) u^{2}$ .

$3 \sqrt{27} \int_{1}^{\frac{2 \sqrt{3}}{3}} - 1 u^{2} + u^{2} u^{2} d u$

Step 10

Simplify.

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Step 10.1

Rewrite $- 1 u^{2}$ as $- u^{2}$ .

$3 \sqrt{27} \int_{1}^{\frac{2 \sqrt{3}}{3}} - u^{2} + u^{2} u^{2} d u$

Step 10.2

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 10.2.1

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

$3 \sqrt{27} \int_{1}^{\frac{2 \sqrt{3}}{3}} - u^{2} + u^{2 + 2} d u$

Step 10.2.2

Add $2$ and $2$ .

$3 \sqrt{27} \int_{1}^{\frac{2 \sqrt{3}}{3}} - u^{2} + u^{4} d u$

Step 11

Split the single integral into multiple integrals.

$3 \sqrt{27} (\int_{1}^{\frac{2 \sqrt{3}}{3}} - u^{2} d u + \int_{1}^{\frac{2 \sqrt{3}}{3}} u^{4} d u)$

Step 12

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$3 \sqrt{27} (- \int_{1}^{\frac{2 \sqrt{3}}{3}} u^{2} d u + \int_{1}^{\frac{2 \sqrt{3}}{3}} u^{4} d u)$

Step 13

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$3 \sqrt{27} (- (\frac{1}{3} u^{3}]_{1}^{\frac{2 \sqrt{3}}{3}}) + \int_{1}^{\frac{2 \sqrt{3}}{3}} u^{4} d u)$

Step 14

Combine $\frac{1}{3}$ and $u^{3}$ .

$3 \sqrt{27} (- (\frac{u^{3}}{3}]_{1}^{\frac{2 \sqrt{3}}{3}}) + \int_{1}^{\frac{2 \sqrt{3}}{3}} u^{4} d u)$

Step 15

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$3 \sqrt{27} (- (\frac{u^{3}}{3}]_{1}^{\frac{2 \sqrt{3}}{3}}) + \frac{1}{5} u^{5}]_{1}^{\frac{2 \sqrt{3}}{3}})$

Step 16

Combine $\frac{1}{5}$ and $u^{5}$ .

$3 \sqrt{27} (- (\frac{u^{3}}{3}]_{1}^{\frac{2 \sqrt{3}}{3}}) + \frac{u^{5}}{5}]_{1}^{\frac{2 \sqrt{3}}{3}})$

Step 17

Substitute and simplify.

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Step 17.1

Evaluate $\frac{u^{3}}{3}$ at $\frac{2 \sqrt{3}}{3}$ and at $1$ .

$3 \sqrt{27} (- ((\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3}) - \frac{1^{3}}{3}) + \frac{u^{5}}{5}]_{1}^{\frac{2 \sqrt{3}}{3}})$

Step 17.2

Evaluate $\frac{u^{5}}{5}$ at $\frac{2 \sqrt{3}}{3}$ and at $1$ .

$3 \sqrt{27} (- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1^{3}}{3}) + \frac{{(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1^{5}}{5})$

Step 17.3

Simplify.

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Step 17.3.1

One to any power is one.

$3 \sqrt{27} (- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + \frac{{(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1^{5}}{5})$

Step 17.3.2

One to any power is one.

$3 \sqrt{27} (- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + \frac{{(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 17.3.3

To write $- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3})$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3 \sqrt{27} (- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) \cdot \frac{5}{5} + \frac{{(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 17.3.4

Combine $- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3})$ and $\frac{5}{5}$ .

$3 \sqrt{27} (\frac{- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) \cdot 5}{5} + \frac{{(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 17.3.5

Combine the numerators over the common denominator.

$3 \sqrt{27} (\frac{- (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) \cdot 5 + {(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 17.3.6

Multiply $5$ by $- 1$ .

$3 \sqrt{27} (\frac{- 5 (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 18

Simplify.

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Step 18.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 18.1.1

Factor $9$ out of $27$ .

$3 \sqrt{9 (3)} (\frac{- 5 (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 18.1.2

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

$3 \sqrt{3^{2} \cdot 3} (\frac{- 5 (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 18.2

Pull terms out from under the radical.

$3 (3 \sqrt{3}) (\frac{- 5 (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 18.3

Multiply $3$ by $3$ .

$9 \sqrt{3} (\frac{- 5 (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5}}{5} - \frac{1}{5})$

Step 19

Simplify.

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Step 19.1

Combine the numerators over the common denominator.

$9 \sqrt{3} \frac{- 5 (\frac{{(\frac{2 \sqrt{3}}{3})}^{3}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2

Simplify each term.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Simplify the numerator.

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Step 19.2.1.1.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{{(2 \sqrt{3})}^{3}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2

Simplify the numerator.

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Step 19.2.1.1.2.1

Apply the product rule to $2 \sqrt{3}$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{2^{3} {\sqrt{3}}^{3}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.2

Raise $2$ to the power of $3$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 {\sqrt{3}}^{3}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.3

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 \sqrt{3^{3}}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.4

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

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 \sqrt{27}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.5

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 19.2.1.1.2.5.1

Factor $9$ out of $27$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 \sqrt{9 (3)}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.5.2

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

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.6

Pull terms out from under the radical.

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 \cdot 3 \sqrt{3}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.2.7

Multiply $8$ by $3$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{24 \sqrt{3}}{3^{3}}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.3

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

$9 \sqrt{3} \frac{- 5 (\frac{\frac{24 \sqrt{3}}{27}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.4

Cancel the common factor of $24$ and $27$ .

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Step 19.2.1.1.4.1

Factor $3$ out of $24 \sqrt{3}$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{3 (8 \sqrt{3})}{27}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.4.2

Cancel the common factors.

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Step 19.2.1.1.4.2.1

Factor $3$ out of $27$ .

$9 \sqrt{3} \frac{- 5 (\frac{\frac{3 (8 \sqrt{3})}{3 (9)}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.4.2.2

Cancel the common factor.

$9 \sqrt{3} \frac{- 5 (\frac{\frac{3 (8 \sqrt{3})}{3 \cdot 9}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.1.4.2.3

Rewrite the expression.

$9 \sqrt{3} \frac{- 5 (\frac{\frac{8 \sqrt{3}}{9}}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$9 \sqrt{3} \frac{- 5 (\frac{8 \sqrt{3}}{9} \cdot \frac{1}{3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.3

Multiply $\frac{8 \sqrt{3}}{9} \cdot \frac{1}{3}$ .

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Step 19.2.1.3.1

Multiply $\frac{8 \sqrt{3}}{9}$ by $\frac{1}{3}$ .

$9 \sqrt{3} \frac{- 5 (\frac{8 \sqrt{3}}{9 \cdot 3} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.1.3.2

Multiply $9$ by $3$ .

$9 \sqrt{3} \frac{- 5 (\frac{8 \sqrt{3}}{27} - \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.2

Apply the distributive property.

$9 \sqrt{3} \frac{- 5 \frac{8 \sqrt{3}}{27} - 5 (- \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.3

Multiply $- 5 \frac{8 \sqrt{3}}{27}$ .

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Step 19.2.3.1

Combine $- 5$ and $\frac{8 \sqrt{3}}{27}$ .

$9 \sqrt{3} \frac{\frac{- 5 (8 \sqrt{3})}{27} - 5 (- \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.3.2

Multiply $8$ by $- 5$ .

$9 \sqrt{3} \frac{\frac{- 40 \sqrt{3}}{27} - 5 (- \frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.4

Multiply $- 5 (- \frac{1}{3})$ .

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Step 19.2.4.1

Multiply $- 1$ by $- 5$ .

$9 \sqrt{3} \frac{\frac{- 40 \sqrt{3}}{27} + 5 (\frac{1}{3}) + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.4.2

Combine $5$ and $\frac{1}{3}$ .

$9 \sqrt{3} \frac{\frac{- 40 \sqrt{3}}{27} + \frac{5}{3} + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.5

Move the negative in front of the fraction.

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + {(\frac{2 \sqrt{3}}{3})}^{5} - 1}{5}$

Step 19.2.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 19.2.6.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{{(2 \sqrt{3})}^{5}}{3^{5}} - 1}{5}$

Step 19.2.6.2

Apply the product rule to $2 \sqrt{3}$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{2^{5} {\sqrt{3}}^{5}}{3^{5}} - 1}{5}$

Step 19.2.7

Simplify the numerator.

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Step 19.2.7.1

Raise $2$ to the power of $5$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 {\sqrt{3}}^{5}}{3^{5}} - 1}{5}$

Step 19.2.7.2

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 \sqrt{3^{5}}}{3^{5}} - 1}{5}$

Step 19.2.7.3

Raise $3$ to the power of $5$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 \sqrt{243}}{3^{5}} - 1}{5}$

Step 19.2.7.4

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 19.2.7.4.1

Factor $81$ out of $243$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 \sqrt{81 (3)}}{3^{5}} - 1}{5}$

Step 19.2.7.4.2

Rewrite $81$ as $9^{2}$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 \sqrt{9^{2} \cdot 3}}{3^{5}} - 1}{5}$

Step 19.2.7.5

Pull terms out from under the radical.

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 \cdot 9 \sqrt{3}}{3^{5}} - 1}{5}$

Step 19.2.7.6

Multiply $32$ by $9$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{288 \sqrt{3}}{3^{5}} - 1}{5}$

Step 19.2.8

Raise $3$ to the power of $5$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{288 \sqrt{3}}{243} - 1}{5}$

Step 19.2.9

Cancel the common factor of $288$ and $243$ .

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Step 19.2.9.1

Factor $9$ out of $288 \sqrt{3}$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{9 (32 \sqrt{3})}{243} - 1}{5}$

Step 19.2.9.2

Cancel the common factors.

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Step 19.2.9.2.1

Factor $9$ out of $243$ .

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{9 (32 \sqrt{3})}{9 (27)} - 1}{5}$

Step 19.2.9.2.2

Cancel the common factor.

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{9 (32 \sqrt{3})}{9 \cdot 27} - 1}{5}$

Step 19.2.9.2.3

Rewrite the expression.

$9 \sqrt{3} \frac{- \frac{40 \sqrt{3}}{27} + \frac{5}{3} + \frac{32 \sqrt{3}}{27} - 1}{5}$

Step 19.3

Combine the numerators over the common denominator.

$9 \sqrt{3} \frac{- 1 + \frac{- 40 \sqrt{3} + 32 \sqrt{3}}{27} + \frac{5}{3}}{5}$

Step 19.4

Add $- 40 \sqrt{3}$ and $32 \sqrt{3}$ .

$9 \sqrt{3} \frac{- 1 + \frac{- 8 \sqrt{3}}{27} + \frac{5}{3}}{5}$

Step 19.5

Move the negative in front of the fraction.

$9 \sqrt{3} \frac{- 1 - \frac{8 \sqrt{3}}{27} + \frac{5}{3}}{5}$

Step 19.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 \sqrt{3} \frac{- 1 \cdot \frac{3}{3} + \frac{5}{3} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.7

Combine $- 1$ and $\frac{3}{3}$ .

$9 \sqrt{3} \frac{\frac{- 1 \cdot 3}{3} + \frac{5}{3} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.8

Combine the numerators over the common denominator.

$9 \sqrt{3} \frac{\frac{- 1 \cdot 3 + 5}{3} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.9

Simplify the numerator.

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Step 19.9.1

Multiply $- 1$ by $3$ .

$9 \sqrt{3} \frac{\frac{- 3 + 5}{3} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.9.2

Add $- 3$ and $5$ .

$9 \sqrt{3} \frac{\frac{2}{3} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.10

Simplify the numerator.

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Step 19.10.1

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$9 \sqrt{3} \frac{\frac{2}{3} \cdot \frac{9}{9} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.10.2

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 19.10.2.1

Multiply $\frac{2}{3}$ by $\frac{9}{9}$ .

$9 \sqrt{3} \frac{\frac{2 \cdot 9}{3 \cdot 9} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.10.2.2

Multiply $3$ by $9$ .

$9 \sqrt{3} \frac{\frac{2 \cdot 9}{27} - \frac{8 \sqrt{3}}{27}}{5}$

Step 19.10.3

Combine the numerators over the common denominator.

$9 \sqrt{3} \frac{\frac{2 \cdot 9 - 8 \sqrt{3}}{27}}{5}$

Step 19.10.4

Multiply $2$ by $9$ .

$9 \sqrt{3} \frac{\frac{18 - 8 \sqrt{3}}{27}}{5}$

Step 19.11

Multiply the numerator by the reciprocal of the denominator.

$9 \sqrt{3} (\frac{18 - 8 \sqrt{3}}{27} \cdot \frac{1}{5})$

Step 19.12

Multiply $\frac{18 - 8 \sqrt{3}}{27} \cdot \frac{1}{5}$ .

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Step 19.12.1

Multiply $\frac{18 - 8 \sqrt{3}}{27}$ by $\frac{1}{5}$ .

$9 \sqrt{3} \frac{18 - 8 \sqrt{3}}{27 \cdot 5}$

Step 19.12.2

Multiply $27$ by $5$ .

$9 \sqrt{3} \frac{18 - 8 \sqrt{3}}{135}$

Step 19.13

Cancel the common factor of $9$ .

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Step 19.13.1

Factor $9$ out of $9 \sqrt{3}$ .

$9 (\sqrt{3}) \frac{18 - 8 \sqrt{3}}{135}$

Step 19.13.2

Factor $9$ out of $135$ .

$9 (\sqrt{3}) \frac{18 - 8 \sqrt{3}}{9 (15)}$

Step 19.13.3

Cancel the common factor.

$9 \sqrt{3} \frac{18 - 8 \sqrt{3}}{9 \cdot 15}$

Step 19.13.4

Rewrite the expression.

$\sqrt{3} \frac{18 - 8 \sqrt{3}}{15}$

Step 19.14

Combine $\sqrt{3}$ and $\frac{18 - 8 \sqrt{3}}{15}$ .

$\frac{\sqrt{3} (18 - 8 \sqrt{3})}{15}$

Step 19.15

Apply the distributive property.

$\frac{\sqrt{3} \cdot 18 + \sqrt{3} (- 8 \sqrt{3})}{15}$

Step 19.16

Move $18$ to the left of $\sqrt{3}$ .

$\frac{18 \cdot \sqrt{3} + \sqrt{3} (- 8 \sqrt{3})}{15}$

Step 19.17

Multiply $\sqrt{3} (- 8 \sqrt{3})$ .

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Step 19.17.1

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{18 \cdot \sqrt{3} - 8 ({\sqrt{3}}^{1} \sqrt{3})}{15}$

Step 19.17.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{18 \cdot \sqrt{3} - 8 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{15}$

Step 19.17.3

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

$\frac{18 \cdot \sqrt{3} - 8 {\sqrt{3}}^{1 + 1}}{15}$

Step 19.17.4

Add $1$ and $1$ .

$\frac{18 \cdot \sqrt{3} - 8 {\sqrt{3}}^{2}}{15}$

Step 19.18

Simplify each term.

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Step 19.18.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 19.18.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{18 \sqrt{3} - 8 {(3^{\frac{1}{2}})}^{2}}{15}$

Step 19.18.1.2

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

$\frac{18 \sqrt{3} - 8 \cdot 3^{\frac{1}{2} \cdot 2}}{15}$

Step 19.18.1.3

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

$\frac{18 \sqrt{3} - 8 \cdot 3^{\frac{2}{2}}}{15}$

Step 19.18.1.4

Cancel the common factor of $2$ .

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Step 19.18.1.4.1

Cancel the common factor.

$\frac{18 \sqrt{3} - 8 \cdot 3^{\frac{2}{2}}}{15}$

Step 19.18.1.4.2

Rewrite the expression.

$\frac{18 \sqrt{3} - 8 \cdot 3^{1}}{15}$

Step 19.18.1.5

Evaluate the exponent.

$\frac{18 \sqrt{3} - 8 \cdot 3}{15}$

Step 19.18.2

Multiply $- 8$ by $3$ .

$\frac{18 \sqrt{3} - 24}{15}$

Step 19.19

Cancel the common factor of $18 \sqrt{3} - 24$ and $15$ .

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Step 19.19.1

Factor $3$ out of $18 \sqrt{3}$ .

$\frac{3 (6 \sqrt{3}) - 24}{15}$

Step 19.19.2

Factor $3$ out of $- 24$ .

$\frac{3 (6 \sqrt{3}) + 3 (- 8)}{15}$

Step 19.19.3

Factor $3$ out of $3 (6 \sqrt{3}) + 3 (- 8)$ .

$\frac{3 (6 \sqrt{3} - 8)}{15}$

Step 19.19.4

Cancel the common factors.

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Step 19.19.4.1

Factor $3$ out of $15$ .

$\frac{3 (6 \sqrt{3} - 8)}{3 (5)}$

Step 19.19.4.2

Cancel the common factor.

$\frac{3 (6 \sqrt{3} - 8)}{3 \cdot 5}$

Step 19.19.4.3

Rewrite the expression.

$\frac{6 \sqrt{3} - 8}{5}$

Step 20

The result can be shown in multiple forms.

Exact Form:

$\frac{6 \sqrt{3} - 8}{5}$

Decimal Form:

$0.47846096 \dots$

Step 21