How do you use the binomial theorem to expand (u^(3/5)+2)^5?

Oct 20, 2017

See below.

Explanation:

$\left(\begin{matrix}5 \\ 0\end{matrix}\right) {\left({u}^{\frac{5}{3}}\right)}^{5} \cdot {2}^{0} + \left(\begin{matrix}5 \\ 1\end{matrix}\right) {\left({u}^{\frac{5}{3}}\right)}^{4} \cdot {2}^{1} + \left(\begin{matrix}5 \\ 2\end{matrix}\right) {\left({u}^{\frac{5}{3}}\right)}^{3} \cdot {2}^{2} \to + \left(\begin{matrix}5 \\ 3\end{matrix}\right) {\left({u}^{\frac{5}{3}}\right)}^{2} \cdot {2}^{3} + \left(\begin{matrix}5 \\ 4\end{matrix}\right) {\left({u}^{\frac{5}{3}}\right)}^{1} \cdot {2}^{4} + \left(\begin{matrix}5 \\ 5\end{matrix}\right) {\left({u}^{\frac{5}{3}}\right)}^{0} \cdot {2}^{5}$

${u}^{\frac{25}{3}} + 10 {u}^{\frac{20}{3}} + 40 {u}^{5} + 80 {u}^{\frac{10}{3}} + 80 {u}^{\frac{5}{3}} + 32$

As can be seen the exponent of the first term decreases from n to 0 and at the same time the exponent of the second term increases from 0 to n.

$\left(\begin{matrix}n \\ r\end{matrix}\right)$ is called the binomial coefficient and is the number of
combinations of n things taken r at a time. This can also be expressed as $n C r$