To simplify the expression
[tex]\[
\frac{\left(x^2\right)^7 \left(x^6\right)^4}{\left(x^2\right)^4},
\][/tex]
we'll use the laws of exponents. Here are the steps:
1. Apply the power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
- For [tex]\(\left(x^2\right)^7\)[/tex]:
[tex]\[
\left(x^2\right)^7 = x^{2 \cdot 7} = x^{14}
\][/tex]
- For [tex]\(\left(x^6\right)^4\)[/tex]:
[tex]\[
\left(x^6\right)^4 = x^{6 \cdot 4} = x^{24}
\][/tex]
- For [tex]\(\left(x^2\right)^4\)[/tex]:
[tex]\[
\left(x^2\right)^4 = x^{2 \cdot 4} = x^8
\][/tex]
2. Simplify the numerator by multiplying powers of [tex]\(x\)[/tex]:
[tex]\[
x^{14} \cdot x^{24}
\][/tex]
Using the multiplication rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we get:
[tex]\[
x^{14+24} = x^{38}
\][/tex]
3. Rewrite the expression using the simplified numerator and denominator:
[tex]\[
\frac{x^{38}}{x^8}
\][/tex]
4. Simplify the fraction using the division rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
x^{38 - 8} = x^{30}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
x^{30}
\][/tex]
So the final answer is:
[tex]\[
x^{30}
\][/tex]
