Expand Using the Binomial Theorem (5+z)^4

${(5 + z)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(5)}^{4 - k} \cdot {(z)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} \cdot {(5)}^{4 - 0} \cdot {(z)}^{0} + \frac{4!}{(4 - 1)! 1!} \cdot {(5)}^{4 - 1} \cdot {(z)}^{1} + \frac{4!}{(4 - 2)! 2!} \cdot {(5)}^{4 - 2} \cdot {(z)}^{2} + \frac{4!}{(4 - 3)! 3!} \cdot {(5)}^{4 - 3} \cdot {(z)}^{3} + \frac{4!}{(4 - 4)! 4!} \cdot {(5)}^{4 - 4} \cdot {(z)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(5)}^{4} \cdot {(z)}^{0} + 4 \cdot {(5)}^{3} \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Step 4

Simplify each term.

Multiply ${(5)}^{4}$ by $1$ .

${(5)}^{4} \cdot {(z)}^{0} + 4 \cdot {(5)}^{3} \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Raise $5$ to the power of $4$ .

$625 \cdot {(z)}^{0} + 4 \cdot {(5)}^{3} \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Anything raised to $0$ is $1$ .

$625 \cdot 1 + 4 \cdot {(5)}^{3} \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Multiply $625$ by $1$ .

$625 + 4 \cdot {(5)}^{3} \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

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

$625 + 4 \cdot 125 \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Multiply $4$ by $125$ .

$625 + 500 \cdot {(z)}^{1} + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Simplify.

$625 + 500 \cdot z + 6 \cdot {(5)}^{2} \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

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

$625 + 500 z + 6 \cdot 25 \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Multiply $6$ by $25$ .

$625 + 500 z + 150 \cdot {(z)}^{2} + 4 \cdot {(5)}^{1} \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Evaluate the exponent.

$625 + 500 z + 150 z^{2} + 4 \cdot 5 \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

Multiply $4$ by $5$ .

$625 + 500 z + 150 z^{2} + 20 \cdot {(z)}^{3} + 1 \cdot {(5)}^{0} \cdot {(z)}^{4}$

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

$625 + 500 z + 150 z^{2} + 20 z^{3} + {(5)}^{0} \cdot {(z)}^{4}$

Anything raised to $0$ is $1$ .

$625 + 500 z + 150 z^{2} + 20 z^{3} + 1 \cdot {(z)}^{4}$

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

$625 + 500 z + 150 z^{2} + 20 z^{3} + z^{4}$

Expand Using the Binomial Theorem (5+z)^4

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