Expand Using the Binomial Theorem (a+2b)^3

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $2 b$ .

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

Multiply $a^{3}$ by $1$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $a^{3}$ by $1$ .

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $2$ .

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

Simplify.

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

Apply the product rule to $2 b$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $3$ by $4$ .

$a^{3} + 6 a^{2} b + 12 a b^{2} + 1 \cdot {(a)}^{0} \cdot {(2 b)}^{3}$

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

$a^{3} + 6 a^{2} b + 12 a b^{2} + {(a)}^{0} \cdot {(2 b)}^{3}$

Anything raised to $0$ is $1$ .

$a^{3} + 6 a^{2} b + 12 a b^{2} + 1 \cdot {(2 b)}^{3}$

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

$a^{3} + 6 a^{2} b + 12 a b^{2} + {(2 b)}^{3}$

Apply the product rule to $2 b$ .

$a^{3} + 6 a^{2} b + 12 a b^{2} + 2^{3} b^{3}$

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

$a^{3} + 6 a^{2} b + 12 a b^{2} + 8 b^{3}$

Expand Using the Binomial Theorem (a+2b)^3

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