why is $1+4e^t + 6e^{2t} + 4e^{3t} +e^{4t} = (1+ e^t)^4$

Question

I have a problem with this question. Thanks for your help.

I'm not sure why I'm not meeting quality standards. Oh well.

Answer 1

Put $x = e^t$

Expand $(1+x)^4$ as a series of powers of $x$ using the binomial theorem.

$(1+x)^4= 1+4x+6x^2+4x^3+x^4$ (if you don't know how I got this, please leave a comment).

Now use $(e^t)^k = e^{kt}$ (rules of exponentiation) to get the required result.

For example $x^2 = (e^t)^2 = e^{2t}$.

Answer 2

As others mentioned, this is the Binomial Theorem. Alternatively \begin{align} 1+4e^t+6e^{2t}+4e^{3t}+e^{4t}&=\color{green}{1}^2+2\cdot\color{green}{1}\cdot\color{blue}{2e^t}+2\cdot\color{green}{1}\cdot\color{red}{e^{2t}}+\left(\color{blue}{2e^t}\right)^2+2\cdot\color{blue}{2e^t}\cdot\color{red}{e^{2t}}+\left(\color{red}{e^{2t}}\right)^2\\ &=\left(\color{green}{1}+\color{blue}{2e^t}+\color{red}{e^{2t}}\right)^2\\ &=\left(\color{violet}{1}+2\cdot\color{violet}{1}\cdot\color{orange}{e^t}+\left(\color{orange}{e^t}\right)^2\right)^2\\ &=\left(\left(\color{violet}{1}+\color{orange}{e^t}\right)^2\right)^2\\ &=\left(1+e^t\right)^4 \end{align}