Calculus Examples

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

Step 1

Use the Binomial Theorem.

$\frac{1}{3} \cdot ({(x^{3})}^{6} + 6 {(x^{3})}^{5} \cdot 2 + 15 {(x^{3})}^{4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply the exponents in ${(x^{3})}^{6}$ .

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Step 2.1.1.1

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

$\frac{1}{3} \cdot (x^{3 \cdot 6} + 6 {(x^{3})}^{5} \cdot 2 + 15 {(x^{3})}^{4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.1.2

Multiply $3$ by $6$ .

$\frac{1}{3} \cdot (x^{18} + 6 {(x^{3})}^{5} \cdot 2 + 15 {(x^{3})}^{4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.2

Multiply the exponents in ${(x^{3})}^{5}$ .

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Step 2.1.2.1

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

$\frac{1}{3} \cdot (x^{18} + 6 x^{3 \cdot 5} \cdot 2 + 15 {(x^{3})}^{4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.2.2

Multiply $3$ by $5$ .

$\frac{1}{3} \cdot (x^{18} + 6 x^{15} \cdot 2 + 15 {(x^{3})}^{4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.3

Multiply $2$ by $6$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 15 {(x^{3})}^{4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.4

Multiply the exponents in ${(x^{3})}^{4}$ .

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Step 2.1.4.1

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

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 15 x^{3 \cdot 4} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.4.2

Multiply $3$ by $4$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 15 x^{12} \cdot 2^{2} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.5

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

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 15 x^{12} \cdot 4 + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.6

Multiply $4$ by $15$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 20 {(x^{3})}^{3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.7

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 2.1.7.1

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

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 20 x^{3 \cdot 3} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.7.2

Multiply $3$ by $3$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 20 x^{9} \cdot 2^{3} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.8

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

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 20 x^{9} \cdot 8 + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.9

Multiply $8$ by $20$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 15 {(x^{3})}^{2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.10

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.1.10.1

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

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 15 x^{3 \cdot 2} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.10.2

Multiply $3$ by $2$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 15 x^{6} \cdot 2^{4} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.11

Raise $2$ to the power of $4$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 15 x^{6} \cdot 16 + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.12

Multiply $16$ by $15$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 240 x^{6} + 6 x^{3} \cdot 2^{5} + 2^{6})$

Step 2.1.13

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

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 240 x^{6} + 6 x^{3} \cdot 32 + 2^{6})$

Step 2.1.14

Multiply $32$ by $6$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 240 x^{6} + 192 x^{3} + 2^{6})$

Step 2.1.15

Raise $2$ to the power of $6$ .

$\frac{1}{3} \cdot (x^{18} + 12 x^{15} + 60 x^{12} + 160 x^{9} + 240 x^{6} + 192 x^{3} + 64)$

Step 2.2

Apply the distributive property.

$\frac{1}{3} x^{18} + \frac{1}{3} (12 x^{15}) + \frac{1}{3} (60 x^{12}) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{3}$ and $x^{18}$ .

$\frac{x^{18}}{3} + \frac{1}{3} (12 x^{15}) + \frac{1}{3} (60 x^{12}) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Factor $3$ out of $12 x^{15}$ .

$\frac{x^{18}}{3} + \frac{1}{3} (3 (4 x^{15})) + \frac{1}{3} (60 x^{12}) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.2.2

Cancel the common factor.

$\frac{x^{18}}{3} + \frac{1}{3} (3 (4 x^{15})) + \frac{1}{3} (60 x^{12}) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.2.3

Rewrite the expression.

$\frac{x^{18}}{3} + 4 x^{15} + \frac{1}{3} (60 x^{12}) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Factor $3$ out of $60 x^{12}$ .

$\frac{x^{18}}{3} + 4 x^{15} + \frac{1}{3} (3 (20 x^{12})) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.3.2

Cancel the common factor.

$\frac{x^{18}}{3} + 4 x^{15} + \frac{1}{3} (3 (20 x^{12})) + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.3.3

Rewrite the expression.

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{1}{3} (160 x^{9}) + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.4

Multiply $\frac{1}{3} (160 x^{9})$ .

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Step 3.4.1

Combine $160$ and $\frac{1}{3}$ .

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160}{3} x^{9} + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.4.2

Combine $\frac{160}{3}$ and $x^{9}$ .

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + \frac{1}{3} (240 x^{6}) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.5

Cancel the common factor of $3$ .

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Step 3.5.1

Factor $3$ out of $240 x^{6}$ .

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + \frac{1}{3} (3 (80 x^{6})) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.5.2

Cancel the common factor.

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + \frac{1}{3} (3 (80 x^{6})) + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.5.3

Rewrite the expression.

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + 80 x^{6} + \frac{1}{3} (192 x^{3}) + \frac{1}{3} \cdot 64$

Step 3.6

Cancel the common factor of $3$ .

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Step 3.6.1

Factor $3$ out of $192 x^{3}$ .

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + 80 x^{6} + \frac{1}{3} (3 (64 x^{3})) + \frac{1}{3} \cdot 64$

Step 3.6.2

Cancel the common factor.

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + 80 x^{6} + \frac{1}{3} (3 (64 x^{3})) + \frac{1}{3} \cdot 64$

Step 3.6.3

Rewrite the expression.

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + 80 x^{6} + 64 x^{3} + \frac{1}{3} \cdot 64$

Step 3.7

Combine $\frac{1}{3}$ and $64$ .

$\frac{x^{18}}{3} + 4 x^{15} + 20 x^{12} + \frac{160 x^{9}}{3} + 80 x^{6} + 64 x^{3} + \frac{64}{3}$