Learn how to solve problems step by step online. Expand the expression e^(3x)(e^(3x)-8)^5. We can expand the expression \left(e^{3x}-8\right)^5 using Newton's binomial theorem, which is a formula that allow us to find the expanded form of a binomial raised to a positive integer n. The formula is as follows: \displaystyle(a\pm b)^n=\sum_{k=0}^{n}\left(\begin{matrix}n\\k\end{matrix}\right)a^{n-k}b^k=\left(\begin{matrix}n\\0\end{matrix}\right)a^n\pm\left(\begin{matrix}n\\1\end{matrix}\right)a^{n-1}b+\left(\begin{matrix}n\\2\end{matrix}\right)a^{n-2}b^2\pm\dots\pm\left(\begin{matrix}n\\n\end{matrix}\right)b^n. The number of terms resulting from the expansion always equals n + 1. The coefficients \left(\begin{matrix}n\\k\end{matrix}\right) are combinatorial numbers which correspond to the nth row of the Tartaglia triangle (or Pascal's triangle). In the formula, we can observe that the exponent of a decreases, from n to 0, while the exponent of b increases, from 0 to n. If one of the binomial terms is negative, the positive and negative signs alternate.. Multiply the single term e^{3x} by each term of the polynomial \left(e^{15x}-40e^{12x}+640e^{9x}-5120e^{6x}+20480e^{3x}-32768\right). When multiplying exponents with same base we can add the exponents. When multiplying two powers that have the same base (e^{3x}), you can add the exponents.

Expand the expression e^(3x)(e^(3x)-8)^5