Expand Using the Binomial Theorem (n+5)^2

${(n + 5)}^{2}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

Multiply $n^{2}$ by $1$ .

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

Simplify.

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

Evaluate the exponent.

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

Multiply $5$ by $2$ .

$n^{2} + 10 n + 1 \cdot {(n)}^{0} \cdot {(5)}^{2}$

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

$n^{2} + 10 n + {(n)}^{0} \cdot {(5)}^{2}$

Anything raised to $0$ is $1$ .

$n^{2} + 10 n + 1 \cdot {(5)}^{2}$

Multiply ${(5)}^{2}$ by $1$ .

$n^{2} + 10 n + {(5)}^{2}$

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

$n^{2} + 10 n + 25$

Expand Using the Binomial Theorem (n+5)^2

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