Calculus Examples

$V (r) = (\frac{4}{3}) p r^{3}$ , $V (r + 3)$

Step 1

Replace the variable $r$ with $r + 3$ in the expression.

$V (r + 3) = (\frac{4}{3}) \cdot (p {(r + 3)}^{3})$

Step 2

Simplify $(\frac{4}{3}) \cdot (p {(r + 3)}^{3})$ .

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Step 2.1

Remove parentheses.

$(\frac{4}{3}) \cdot (p {(r + 3)}^{3})$

Step 2.2

Use the Binomial Theorem.

$\frac{4}{3} \cdot (p (r^{3} + 3 r^{2} \cdot 3 + 3 r \cdot 3^{2} + 3^{3}))$

Step 2.3

Simplify terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $3$ by $3$ .

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 3 r \cdot 3^{2} + 3^{3}))$

Step 2.3.1.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 2.3.1.2.1

Move $3^{2}$ .

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 3^{2} \cdot 3 r + 3^{3}))$

Step 2.3.1.2.2

Multiply $3^{2}$ by $3$ .

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Step 2.3.1.2.2.1

Raise $3$ to the power of $1$ .

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 3^{2} \cdot 3^{1} r + 3^{3}))$

Step 2.3.1.2.2.2

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

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 3^{2 + 1} r + 3^{3}))$

Step 2.3.1.2.3

Add $2$ and $1$ .

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 3^{3} r + 3^{3}))$

Step 2.3.1.3

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

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 27 r + 3^{3}))$

Step 2.3.1.4

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

$\frac{4}{3} \cdot (p (r^{3} + 9 r^{2} + 27 r + 27))$

Step 2.3.2

Apply the distributive property.

$\frac{4}{3} \cdot (p r^{3} + p (9 r^{2}) + p (27 r) + p \cdot 27)$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite using the commutative property of multiplication.

$\frac{4}{3} \cdot (p r^{3} + 9 p r^{2} + p (27 r) + p \cdot 27)$

Step 2.4.2

Rewrite using the commutative property of multiplication.

$\frac{4}{3} \cdot (p r^{3} + 9 p r^{2} + 27 p r + p \cdot 27)$

Step 2.4.3

Move $27$ to the left of $p$ .

$\frac{4}{3} \cdot (p r^{3} + 9 p r^{2} + 27 p r + 27 \cdot p)$

Step 2.5

Apply the distributive property.

$\frac{4}{3} (p r^{3}) + \frac{4}{3} (9 p r^{2}) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6

Simplify.

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Step 2.6.1

Multiply $\frac{4}{3} (p r^{3})$ .

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Step 2.6.1.1

Combine $p$ and $\frac{4}{3}$ .

$\frac{p \cdot 4}{3} r^{3} + \frac{4}{3} (9 p r^{2}) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6.1.2

Combine $\frac{p \cdot 4}{3}$ and $r^{3}$ .

$\frac{p \cdot 4 r^{3}}{3} + \frac{4}{3} (9 p r^{2}) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6.2

Cancel the common factor of $3$ .

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Step 2.6.2.1

Factor $3$ out of $9 p r^{2}$ .

$\frac{p \cdot 4 r^{3}}{3} + \frac{4}{3} (3 (3 p r^{2})) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6.2.2

Cancel the common factor.

$\frac{p \cdot 4 r^{3}}{3} + \frac{4}{3} (3 (3 p r^{2})) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6.2.3

Rewrite the expression.

$\frac{p \cdot 4 r^{3}}{3} + 4 (3 p r^{2}) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6.3

Multiply $3$ by $4$ .

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + \frac{4}{3} (27 p r) + \frac{4}{3} (27 p)$

Step 2.6.4

Cancel the common factor of $3$ .

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Step 2.6.4.1

Factor $3$ out of $27 p r$ .

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + \frac{4}{3} (3 (9 p r)) + \frac{4}{3} (27 p)$

Step 2.6.4.2

Cancel the common factor.

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + \frac{4}{3} (3 (9 p r)) + \frac{4}{3} (27 p)$

Step 2.6.4.3

Rewrite the expression.

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + 4 (9 p r) + \frac{4}{3} (27 p)$

Step 2.6.5

Multiply $9$ by $4$ .

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + 36 (p r) + \frac{4}{3} (27 p)$

Step 2.6.6

Cancel the common factor of $3$ .

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Step 2.6.6.1

Factor $3$ out of $27 p$ .

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + 36 (p r) + \frac{4}{3} (3 (9 p))$

Step 2.6.6.2

Cancel the common factor.

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + 36 (p r) + \frac{4}{3} (3 (9 p))$

Step 2.6.6.3

Rewrite the expression.

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + 36 (p r) + 4 (9 p)$

Step 2.6.7

Multiply $9$ by $4$ .

$\frac{p \cdot 4 r^{3}}{3} + 12 (p r^{2}) + 36 (p r) + 36 p$

Step 2.7

Move $4$ to the left of $p$ .

$\frac{4 p r^{3}}{3} + 12 p r^{2} + 36 p r + 36 p$