Sure! Let's simplify the expression step by step:
1. Identify the given expression:
[tex]\[
\frac{x^7 (2x)^5}{x^3}
\][/tex]
2. Simplify the expression inside the numerator:
[tex]\[
(2x)^5
\][/tex]
This can be expanded using the properties of exponents:
[tex]\[
(2x)^5 = 2^5 \cdot x^5
\][/tex]
We know that:
[tex]\[
2^5 = 32
\][/tex]
So:
[tex]\[
(2x)^5 = 32x^5
\][/tex]
3. Substitute the simplified form back into the original expression:
[tex]\[
\frac{x^7 \cdot 32x^5}{x^3}
\][/tex]
4. Combine the terms in the numerator:
[tex]\[
x^7 \cdot 32x^5 = 32 \cdot x^{7+5} = 32x^{12}
\][/tex]
So our expression now is:
[tex]\[
\frac{32x^{12}}{x^3}
\][/tex]
5. Simplify the fraction by dividing exponents:
Use the property of exponents: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{32x^{12}}{x^3} = 32 \cdot x^{12-3} = 32x^9
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{32x^9}
\][/tex]
