Algebra Examples

${(3 p + 4 q)}^{3}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(3 p)}^{3 - k} \cdot {(4 q)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} \cdot {(3 p)}^{3 - 0} \cdot {(4 q)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(3 p)}^{3 - 1} \cdot {(4 q)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(3 p)}^{3 - 2} \cdot {(4 q)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(3 p)}^{3 - 3} \cdot {(4 q)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3 p)}^{3} \cdot {(4 q)}^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(3 p)}^{3}$ by $1$ .

${(3 p)}^{3} \cdot {(4 q)}^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.2

Apply the product rule to $3 p$ .

$3^{3} p^{3} \cdot {(4 q)}^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.3

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

$27 p^{3} \cdot {(4 q)}^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.4

Apply the product rule to $4 q$ .

$27 p^{3} \cdot (4^{0} q^{0}) + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.5

Rewrite using the commutative property of multiplication.

$27 \cdot 4^{0} p^{3} q^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.6

Anything raised to $0$ is $1$ .

$27 \cdot 1 p^{3} q^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.7

Multiply $27$ by $1$ .

$27 p^{3} q^{0} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.8

Anything raised to $0$ is $1$ .

$27 p^{3} \cdot 1 + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.9

Multiply $27$ by $1$ .

$27 p^{3} + 3 \cdot {(3 p)}^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.10

Apply the product rule to $3 p$ .

$27 p^{3} + 3 \cdot (3^{2} p^{2}) \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.11

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

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Step 4.11.1

Move $3^{2}$ .

$27 p^{3} + 3^{2} \cdot 3 \cdot p^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.11.2

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

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Step 4.11.2.1

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

$27 p^{3} + 3^{2} \cdot 3^{1} \cdot p^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.11.2.2

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

$27 p^{3} + 3^{2 + 1} \cdot p^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.11.3

Add $2$ and $1$ .

$27 p^{3} + 3^{3} \cdot p^{2} \cdot {(4 q)}^{1} + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.12

Simplify $3^{3} \cdot p^{2} \cdot {(4 q)}^{1}$ .

$27 p^{3} + 3^{3} \cdot p^{2} \cdot (4 q) + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.13

Rewrite using the commutative property of multiplication.

$27 p^{3} + 3^{3} \cdot 4 p^{2} q + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.14

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

$27 p^{3} + 27 \cdot 4 p^{2} q + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.15

Multiply $27$ by $4$ .

$27 p^{3} + 108 p^{2} q + 3 \cdot {(3 p)}^{1} \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.16

Simplify.

$27 p^{3} + 108 p^{2} q + 3 \cdot (3 p) \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.17

Multiply $3$ by $3$ .

$27 p^{3} + 108 p^{2} q + 9 p \cdot {(4 q)}^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.18

Apply the product rule to $4 q$ .

$27 p^{3} + 108 p^{2} q + 9 p \cdot (4^{2} q^{2}) + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.19

Rewrite using the commutative property of multiplication.

$27 p^{3} + 108 p^{2} q + 9 \cdot 4^{2} p q^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.20

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

$27 p^{3} + 108 p^{2} q + 9 \cdot 16 p q^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.21

Multiply $9$ by $16$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + 1 \cdot {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.22

Multiply ${(3 p)}^{0}$ by $1$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + {(3 p)}^{0} \cdot {(4 q)}^{3}$

Step 4.23

Apply the product rule to $3 p$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + 3^{0} p^{0} \cdot {(4 q)}^{3}$

Step 4.24

Anything raised to $0$ is $1$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + 1 p^{0} \cdot {(4 q)}^{3}$

Step 4.25

Multiply $p^{0}$ by $1$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + p^{0} \cdot {(4 q)}^{3}$

Step 4.26

Anything raised to $0$ is $1$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + 1 \cdot {(4 q)}^{3}$

Step 4.27

Multiply ${(4 q)}^{3}$ by $1$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + {(4 q)}^{3}$

Step 4.28

Apply the product rule to $4 q$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + 4^{3} q^{3}$

Step 4.29

Raise $4$ to the power of $3$ .

$27 p^{3} + 108 p^{2} q + 144 p q^{2} + 64 q^{3}$