Certainly! Let's simplify the given expression step by step using the product rule for exponents.
Given expression:
[tex]\[
\left(-9 m^2 n^4\right)\left(7 m n^2\right)
\][/tex]
We can break this down into three parts: the constants, the [tex]\( m \)[/tex]-terms, and the [tex]\( n \)[/tex]-terms.
1. Simplify the constant terms:
[tex]\[
\left(-9\right) \times \left(7\right) = -63
\][/tex]
2. Simplify the [tex]\( m \)[/tex]-terms:
Use the product rule for exponents which states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex].
[tex]\[
m^2 \times m = m^{2+1} = m^3
\][/tex]
3. Simplify the [tex]\( n \)[/tex]-terms:
Again, use the product rule for exponents:
[tex]\[
n^4 \times n^2 = n^{4+2} = n^6
\][/tex]
Now, combine all the simplified parts together:
[tex]\[
(-63) \times m^3 \times n^6
\][/tex]
So, the expression simplifies to:
[tex]\[
-63m^3n^6
\][/tex]
Thus, the simplified form of the expression using exponents is:
[tex]\[
\boxed{-63m^3n^6}
\][/tex]
