Sure, let's simplify the given expression step-by-step.
The expression to simplify is:
[tex]\[
\frac{w^4\left(w^3\right)^5}{\left(w^5\right)^2}
\][/tex]
1. Simplify the exponentiation inside the parentheses:
- [tex]\((w^3)^5\)[/tex] can be simplified using the power rule [tex]\((a^m)^n = a^{mn}\)[/tex].
[tex]\[
(w^3)^5 = w^{3 \cdot 5} = w^{15}
\][/tex]
- [tex]\((w^5)^2\)[/tex] can also be simplified similarly.
[tex]\[
(w^5)^2 = w^{5 \cdot 2} = w^{10}
\][/tex]
2. Substitute these simplified forms back into the original expression:
[tex]\[
\frac{w^4 \cdot w^{15}}{w^{10}}
\][/tex]
3. Combine the exponents in the numerator using the product rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
w^4 \cdot w^{15} = w^{4+15} = w^{19}
\][/tex]
So the expression now looks like:
[tex]\[
\frac{w^{19}}{w^{10}}
\][/tex]
4. Simplify the fraction using the quotient rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{w^{19}}{w^{10}} = w^{19-10} = w^9
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{w^9}
\][/tex]
