Expand Using the Binomial Theorem (1/2x+4)^10

${(\frac{1}{2} x + 4)}^{10}$

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

$\sum_{k = 0}^{10} \frac{10!}{(10 - k)! k!} \cdot {(\frac{1}{2} x)}^{10 - k} \cdot {(4)}^{k}$

Expand the summation.

$\frac{10!}{(10 - 0)! 0!} \cdot {(\frac{1}{2} x)}^{10 - 0} \cdot {(4)}^{0} + \frac{10!}{(10 - 1)! 1!} \cdot {(\frac{1}{2} x)}^{10 - 1} \cdot {(4)}^{1} + \dots + \frac{10!}{(10 - 10)! 10!} \cdot {(\frac{1}{2} x)}^{10 - 10} \cdot {(4)}^{10}$

Simplify the exponents for each term of the expansion.

$1 \cdot {(\frac{1}{2} x)}^{10} \cdot {(4)}^{0} + 10 \cdot {(\frac{1}{2} x)}^{9} \cdot {(4)}^{1} + \dots + 1 \cdot {(\frac{1}{2} x)}^{0} \cdot {(4)}^{10}$

Simplify the polynomial result.

$\frac{x^{10}}{1024} + \frac{5 x^{9}}{64} + \frac{45 x^{8}}{16} + 60 x^{7} + 840 x^{6} + 8064 x^{5} + 53760 x^{4} + 245760 x^{3} + 737280 x^{2} + 1310720 x + 1048576$

Expand Using the Binomial Theorem (1/2x+4)^10

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