Chapter 11, Problem 13QC

To determine

To expand: The power of expression ${\left(\frac{a}{2}+2\right)}^5$ .

Answer to Problem 13QC

The expansion is $\frac{a^5}{32}+\frac{5a^4}{8}+5a^3+20a^2+40a+32$ .

Explanation of Solution

Given:

The expression is ${\left(\frac{a}{2}+2\right)}^5$ .

Calculation:

If $n$ is natural number, then ${\left(a+b\right)}^n{=}_n{C}_0{a}^n{b}^0{+}_n{C}_1{a}^{n-1}{b}^1{+}_n{C}_2{a}^{n-2}{b}^2+\cdots {+}_n{C}_n{a}^0{b}^n$ .

In this problem $n=5$ , $a=\frac{a}{2}$ and $b=2$ then,

  $\begin{array}{c}{\left(\frac{a}{2}+2\right)}^5=\left[\begin{array}{l}{}_5{C}_0{\left(\frac{a}{2}\right)}^5{\left(2\right)}^0{+}_5{C}_1{\left(\frac{a}{2}\right)}^4{\left(2\right)}^1{+}_5{C}_2{\left(\frac{a}{2}\right)}^3{\left(2\right)}^2{+}_5{C}_3{\left(\frac{a}{2}\right)}^2{\left(2\right)}^3{+}_5{C}_4{\left(\frac{a}{2}\right)}^1{\left(2\right)}^4\\ {+}_5{C}_5{\left(\frac{a}{2}\right)}^0{\left(2\right)}^5\end{array}\right]\\ =\frac{a^5}{32}+\frac{5a^4}{8}+5a^3+20a^2+40a+32\end{array}$

Therefore, the expansion is $\frac{a^5}{32}+\frac{5a^4}{8}+5a^3+20a^2+40a+32$ .