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Chapter 8: Problem 47

Find the term indicated in each expansion. \(\left(x^{2}+y\right)^{22} ;\) the term containing \(y^{14}\)

Short Answer

Expert verified
The term in the binomial expansion \((x^{2} + y)^{22}\) containing \(y^{14}\) is \(7315 \cdot x^{16} y^{14}\).

Step by step solution

01

Identify the binomial and the desired term

The binomial is \((x^{2} + y)\) and the power is 22. The term to find is the one containing \(y^{14}\). As \(y\) only appears once in every term of the expansion, this suggests that \(k = 14\).
02

Find the exponent for \(x\)

From the binomial theorem, exponent for \(a (or x^{2} in this case)\) is \(n - k\). For \(n = 22\) and \(k = 14\), the exponent for \(x^{2}\) is \(22 - 14 = 8\). Thus, \(x^{2}\) is raised to the power of 8, giving \(x^{16}\). So the term containing \(y^{14}\) will also contain \(x^{16}\).
03

Find the coefficient of the term

The coefficient of the term is given by \(\binom{n}{k}\). Substitute \(n = 22\) and \(k = 14\) to calculate \(\binom{22}{14}\), which is 7315.
04

Write out the term

With the coefficient and the powers of \(x\) and \(y\), the term containing \(y^{14}\) in \((x^{2} + y)^{22}\) is \(7315 \cdot x^{16} y^{14}\).

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