Expand Using the Binomial Theorem ( square root of x+ square root of 3)^8

${(\sqrt{x} + \sqrt{3})}^{8}$

Step 1

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

$\sum_{k = 0}^{8} \frac{8!}{(8 - k)! k!} \cdot {(\sqrt{x})}^{8 - k} \cdot {(\sqrt{3})}^{k}$

Step 2

Expand the summation.

$\frac{8!}{(8 - 0)! 0!} \cdot {(\sqrt{x})}^{8 - 0} \cdot {(\sqrt{3})}^{0} + \frac{8!}{(8 - 1)! 1!} \cdot {(\sqrt{x})}^{8 - 1} \cdot {(\sqrt{3})}^{1} + \dots + \frac{8!}{(8 - 8)! 8!} \cdot {(\sqrt{x})}^{8 - 8} \cdot {(\sqrt{3})}^{8}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(\sqrt{x})}^{8} \cdot {(\sqrt{3})}^{0} + 8 \cdot {(\sqrt{x})}^{7} \cdot {(\sqrt{3})}^{1} + \dots + 1 \cdot {(\sqrt{x})}^{0} \cdot {(\sqrt{3})}^{8}$

Step 4

Simplify the polynomial result.

$x^{4} + 8 x^{3} \sqrt{3 x} + 84 x^{3} + 168 x^{2} \sqrt{3 x} + 630 x^{2} + 504 x \sqrt{3 x} + 756 x + 216 \sqrt{3 x} + 81$

Expand Using the Binomial Theorem ( square root of x+ square root of 3)^8

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