Calculus Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $6 x$ .

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

Step 2.2

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

$216 x^{3} + 3 {(6 x)}^{2} \frac{y}{2} + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.3

Apply the product rule to $6 x$ .

$216 x^{3} + 3 (6^{2} x^{2}) \frac{y}{2} + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.4

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

$216 x^{3} + 3 (36 x^{2}) \frac{y}{2} + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.5

Multiply $36$ by $3$ .

$216 x^{3} + 108 x^{2} \frac{y}{2} + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.6

Cancel the common factor of $2$ .

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Step 2.6.1

Factor $2$ out of $108 x^{2}$ .

$216 x^{3} + 2 (54 x^{2}) \frac{y}{2} + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.6.2

Cancel the common factor.

$216 x^{3} + 2 (54 x^{2}) \frac{y}{2} + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.6.3

Rewrite the expression.

$216 x^{3} + 54 x^{2} y + 3 (6 x) {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.7

Multiply $6$ by $3$ .

$216 x^{3} + 54 x^{2} y + 18 x {(\frac{y}{2})}^{2} + {(\frac{y}{2})}^{3}$

Step 2.8

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

$216 x^{3} + 54 x^{2} y + 18 x \frac{y^{2}}{2^{2}} + {(\frac{y}{2})}^{3}$

Step 2.9

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

$216 x^{3} + 54 x^{2} y + 18 x \frac{y^{2}}{4} + {(\frac{y}{2})}^{3}$

Step 2.10

Cancel the common factor of $2$ .

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Step 2.10.1

Factor $2$ out of $18 x$ .

$216 x^{3} + 54 x^{2} y + 2 (9 x) \frac{y^{2}}{4} + {(\frac{y}{2})}^{3}$

Step 2.10.2

Factor $2$ out of $4$ .

$216 x^{3} + 54 x^{2} y + 2 (9 x) \frac{y^{2}}{2 (2)} + {(\frac{y}{2})}^{3}$

Step 2.10.3

Cancel the common factor.

$216 x^{3} + 54 x^{2} y + 2 (9 x) \frac{y^{2}}{2 \cdot 2} + {(\frac{y}{2})}^{3}$

Step 2.10.4

Rewrite the expression.

$216 x^{3} + 54 x^{2} y + 9 x \frac{y^{2}}{2} + {(\frac{y}{2})}^{3}$

Step 2.11

Combine $9$ and $\frac{y^{2}}{2}$ .

$216 x^{3} + 54 x^{2} y + x \frac{9 y^{2}}{2} + {(\frac{y}{2})}^{3}$

Step 2.12

Combine $x$ and $\frac{9 y^{2}}{2}$ .

$216 x^{3} + 54 x^{2} y + \frac{x (9 y^{2})}{2} + {(\frac{y}{2})}^{3}$

Step 2.13

Move $9$ to the left of $x$ .

$216 x^{3} + 54 x^{2} y + \frac{9 \cdot x y^{2}}{2} + {(\frac{y}{2})}^{3}$

Step 2.14

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

$216 x^{3} + 54 x^{2} y + \frac{9 x y^{2}}{2} + \frac{y^{3}}{2^{3}}$

Step 2.15

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

$216 x^{3} + 54 x^{2} y + \frac{9 x y^{2}}{2} + \frac{y^{3}}{8}$