Guided Example #1 - Binomial Theorem

Guided Example

Find the fourth term in the expansion of ${(x^{2} + \sqrt[3]{y})}^{10}$ .

a. $210 x^{6} y^{4}$
b. $210 x^{14} y$
c. $120 x^{6} y^{4}$
d. $120 x^{14} y$

The correct answer is D. To see why, view each of the steps below.

Step 1

Let $a = x^{2}$ and $b = y^{1 ⁄ 3}$ . The exponent of b in the fourth term is $k = 4 - 1 = 3$ ,

Step 2

and the exponent of a is $n - k = 10 - 3 = 7$ . From the binomial theorem we obtain
$(\frac{10}{3}) {(x^{2})}^{7} {(y^{\frac{1}{3}})}^{3} = 120 x^{14} y$ .

All Steps

Let $a = x^{2}$ and $b = y^{1 ⁄ 3}$ . The exponent of b in the fourth term is $k = 4 - 1 = 3$ ,

and the exponent of a is $n - k = 10 - 3 = 7$ . From the binomial theorem we obtain
$(\frac{10}{3}) {(x^{2})}^{7} {(y^{\frac{1}{3}})}^{3} = 120 x^{14} y$ .