Question

homework 6.2 multiplication of monomials
simplify each expression.

1. \\((3m^7)(m^{2n})(5m^3n^8)^2\\)

Explanation:

Step1: Simplify the squared term

First, we simplify \((5m^{3}n^{8})^{2}\) using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{mn}\).
\[
(5m^{3}n^{8})^{2}=5^{2}(m^{3})^{2}(n^{8})^{2}=25m^{6}n^{16}
\]

Step2: Multiply the monomials

Now we multiply the three monomials: \((3m^{7})\), \((m^{2n})\), and \(25m^{6}n^{16}\). First, multiply the coefficients: \(3\times25 = 75\). Then, for the \(m\) terms, use the product of powers rule \(a^m\times a^n=a^{m + n}\): \(m^{7}\times m^{2n}\times m^{6}=m^{7 + 2n+6}=m^{13 + 2n}\). The \(n\) term remains \(n^{16}\) (since there is only one \(n\) term from the third monomial and the first two have no \(n\) terms other than what is in the second monomial, but the second monomial has \(m^{2n}\) which is an \(m\) term, so the \(n\) comes only from the third monomial).
Putting it all together, we get:
\[
(3m^{7})(m^{2n})(25m^{6}n^{16})=75m^{13 + 2n}n^{16}
\]

Answer:

\(75m^{2n + 13}n^{16}\) (or \(75n^{16}m^{2n + 13}\))