Expand Using the Binomial Theorem (x^(1/2)+1)^20

${(x^{\frac{1}{2}} + 1)}^{20}$

Step 1

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

$\sum_{k = 0}^{20} \frac{20!}{(20 - k)! k!} \cdot {(x^{\frac{1}{2}})}^{20 - k} \cdot {(1)}^{k}$

Step 2

Expand the summation.

$\frac{20!}{(20 - 0)! 0!} \cdot {(x^{\frac{1}{2}})}^{20 - 0} \cdot {(1)}^{0} + \frac{20!}{(20 - 1)! 1!} \cdot {(x^{\frac{1}{2}})}^{20 - 1} \cdot {(1)}^{1} + \dots + \frac{20!}{(20 - 20)! 20!} \cdot {(x^{\frac{1}{2}})}^{20 - 20} \cdot {(1)}^{20}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x^{\frac{1}{2}})}^{20} \cdot {(1)}^{0} + 20 \cdot {(x^{\frac{1}{2}})}^{19} \cdot {(1)}^{1} + \dots + 1 \cdot {(x^{\frac{1}{2}})}^{0} \cdot {(1)}^{20}$

Step 4

Simplify the polynomial result.

$x^{10} + 20 x^{\frac{19}{2}} + 190 x^{9} + 1140 x^{\frac{17}{2}} + 4845 x^{8} + 15504 x^{\frac{15}{2}} + 38760 x^{7} + 77520 x^{\frac{13}{2}} + 125970 x^{6} + 167960 x^{\frac{11}{2}} + 184756 x^{5} + 167960 x^{\frac{9}{2}} + 125970 x^{4} + 77520 x^{\frac{7}{2}} + 38760 x^{3} + 15504 x^{\frac{5}{2}} + 4845 x^{2} + 1140 x^{\frac{3}{2}} + 190 x + 20 x^{\frac{1}{2}} + 1$

Expand Using the Binomial Theorem (x^(1/2)+1)^20

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