To solve the expression [tex]\(\frac{9 x^{38}}{6\left(x^6\right)^3 x}\)[/tex], we'll simplify it step-by-step.
1. Simplify the denominator:
[tex]\[
6\left(x^6\right)^3 x
\][/tex]
We need to simplify [tex]\( (x^6)^3 \)[/tex]:
[tex]\[
(x^6)^3 = x^{6 \cdot 3} = x^{18}
\][/tex]
So the denominator becomes:
[tex]\[
6x^{18}x = 6x^{18+1} = 6x^{19}
\][/tex]
2. Write the whole fraction:
[tex]\[
\frac{9 x^{38}}{6 x^{19}}
\][/tex]
3. Simplify the fraction of constants:
[tex]\[
\frac{9}{6} = \frac{3}{2}
\][/tex]
4. Simplify the exponents of [tex]\(x\)[/tex]:
Using the rule [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex], we simplify the exponents:
[tex]\[
\frac{x^{38}}{x^{19}} = x^{38-19} = x^{19}
\][/tex]
5. Combine the simplified constants and exponents:
[tex]\[
\frac{3}{2} x^{19}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{\frac{3}{2} x^{19}}
\][/tex]
