Algebra Examples

${(3 + b)}^{4}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3)}^{4} \cdot {(b)}^{0} + 4 \cdot {(3)}^{3} \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

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

${(3)}^{4} \cdot {(b)}^{0} + 4 \cdot {(3)}^{3} \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.2

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

$81 \cdot {(b)}^{0} + 4 \cdot {(3)}^{3} \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.3

Anything raised to $0$ is $1$ .

$81 \cdot 1 + 4 \cdot {(3)}^{3} \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.4

Multiply $81$ by $1$ .

$81 + 4 \cdot {(3)}^{3} \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.5

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

$81 + 4 \cdot 27 \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.6

Multiply $4$ by $27$ .

$81 + 108 \cdot {(b)}^{1} + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.7

Simplify.

$81 + 108 \cdot b + 6 \cdot {(3)}^{2} \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.8

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

$81 + 108 b + 6 \cdot 9 \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.9

Multiply $6$ by $9$ .

$81 + 108 b + 54 \cdot {(b)}^{2} + 4 \cdot {(3)}^{1} \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.10

Evaluate the exponent.

$81 + 108 b + 54 b^{2} + 4 \cdot 3 \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.11

Multiply $4$ by $3$ .

$81 + 108 b + 54 b^{2} + 12 \cdot {(b)}^{3} + 1 \cdot {(3)}^{0} \cdot {(b)}^{4}$

Step 4.12

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

$81 + 108 b + 54 b^{2} + 12 b^{3} + {(3)}^{0} \cdot {(b)}^{4}$

Step 4.13

Anything raised to $0$ is $1$ .

$81 + 108 b + 54 b^{2} + 12 b^{3} + 1 \cdot {(b)}^{4}$

Step 4.14

Multiply ${(b)}^{4}$ by $1$ .

$81 + 108 b + 54 b^{2} + 12 b^{3} + b^{4}$