Precalculus Examples

${(5 a + 2)}^{3}$

Step 1

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(5 a)}^{3 - k} \cdot {(2)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} \cdot {(5 a)}^{3 - 0} \cdot {(2)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(5 a)}^{3 - 1} \cdot {(2)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(5 a)}^{3 - 2} \cdot {(2)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(5 a)}^{3 - 3} \cdot {(2)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(5 a)}^{3} \cdot {(2)}^{0} + 3 \cdot {(5 a)}^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(5 a)}^{3}$ by $1$ .

${(5 a)}^{3} \cdot {(2)}^{0} + 3 \cdot {(5 a)}^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.2

Apply the product rule to $5 a$ .

$5^{3} a^{3} \cdot {(2)}^{0} + 3 \cdot {(5 a)}^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.3

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

$125 a^{3} \cdot {(2)}^{0} + 3 \cdot {(5 a)}^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.4

Anything raised to $0$ is $1$ .

$125 a^{3} \cdot 1 + 3 \cdot {(5 a)}^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.5

Multiply $125$ by $1$ .

$125 a^{3} + 3 \cdot {(5 a)}^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.6

Apply the product rule to $5 a$ .

$125 a^{3} + 3 \cdot (5^{2} a^{2}) \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.7

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

$125 a^{3} + 3 \cdot (25 a^{2}) \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.8

Multiply $25$ by $3$ .

$125 a^{3} + 75 a^{2} \cdot {(2)}^{1} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.9

Evaluate the exponent.

$125 a^{3} + 75 a^{2} \cdot 2 + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.10

Multiply $2$ by $75$ .

$125 a^{3} + 150 a^{2} + 3 \cdot {(5 a)}^{1} \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.11

Simplify.

$125 a^{3} + 150 a^{2} + 3 \cdot (5 a) \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.12

Multiply $5$ by $3$ .

$125 a^{3} + 150 a^{2} + 15 a \cdot {(2)}^{2} + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.13

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

$125 a^{3} + 150 a^{2} + 15 a \cdot 4 + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.14

Multiply $4$ by $15$ .

$125 a^{3} + 150 a^{2} + 60 a + 1 \cdot {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.15

Multiply ${(5 a)}^{0}$ by $1$ .

$125 a^{3} + 150 a^{2} + 60 a + {(5 a)}^{0} \cdot {(2)}^{3}$

Step 4.16

Apply the product rule to $5 a$ .

$125 a^{3} + 150 a^{2} + 60 a + 5^{0} a^{0} \cdot {(2)}^{3}$

Step 4.17

Anything raised to $0$ is $1$ .

$125 a^{3} + 150 a^{2} + 60 a + 1 a^{0} \cdot {(2)}^{3}$

Step 4.18

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

$125 a^{3} + 150 a^{2} + 60 a + a^{0} \cdot {(2)}^{3}$

Step 4.19

Anything raised to $0$ is $1$ .

$125 a^{3} + 150 a^{2} + 60 a + 1 \cdot {(2)}^{3}$

Step 4.20

Multiply ${(2)}^{3}$ by $1$ .

$125 a^{3} + 150 a^{2} + 60 a + {(2)}^{3}$

Step 4.21

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

$125 a^{3} + 150 a^{2} + 60 a + 8$