Expand Using the Binomial Theorem (a+b)^1

${(a + b)}^{1}$

Step 1

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

$\sum_{k = 0}^{1} \frac{1!}{(1 - k)! k!} \cdot {(a)}^{1 - k} \cdot {(b)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Simplify.

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

Anything raised to $0$ is $1$ .

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

Multiply $a$ by $1$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

$a + {(b)}^{1}$

Simplify.

$a + b$