Okay, let's break down the expression and simplify it step-by-step:
Given expression:
[tex]\[
\frac{\left(4 x^7\right)^{1 / 3}}{(3 x)\left(e^x\right)}
\][/tex]
Step 1: Simplify the numerator [tex]\((4 x^7)^{1/3}\)[/tex].
[tex]\[
(4 x^7)^{1/3} = 4^{1/3} (x^7)^{1/3}
\][/tex]
We know that [tex]\((a^b)^c = a^{bc}\)[/tex], so:
[tex]\[
(x^7)^{1/3} = x^{7 \cdot \frac{1}{3}} = x^{7/3}
\][/tex]
Also, [tex]\(4^{1/3}\)[/tex] is a constant which simplifies as is.
Step 2: Combine the simplified parts.
[tex]\[
4^{1/3} x^{7/3}
\][/tex]
Step 3: Substitute back into the original expression.
[tex]\[
\frac{4^{1/3} x^{7/3}}{3 x \cdot e^x}
\][/tex]
Step 4: Simplify the denominator [tex]\(3 x \cdot e^x\)[/tex].
[tex]\[
3 x e^x
\][/tex]
Step 5: Combine the expressions.
Now divide [tex]\(4^{1/3} x^{7/3}\)[/tex] by [tex]\(3 x e^x\)[/tex]:
[tex]\[
\frac{4^{1/3} x^{7/3}}{3 x e^x}
\][/tex]
Step 6: Simplify the variable terms.
[tex]\[
\frac{4^{1/3} x^{7/3}}{3 x e^x} = \frac{4^{1/3} x^{7/3}}{3 x \cdot e^x} = \frac{4^{1/3} x^{7/3}}{3 x^{1} \cdot e^x}
\][/tex]
Step 7: Subtract the exponents of [tex]\(x\)[/tex].
[tex]\[
\frac{4^{1/3} x^{7/3-1}}{3 e^x}
\][/tex]
[tex]\[
7/3 - 1 = 7/3 - 3/3 = 4/3
\][/tex]
So, we now have:
[tex]\[
\frac{4^{1/3} x^{4/3}}{3 e^x}
\][/tex]
Step 8: Notice that [tex]\(4^{1/3}\)[/tex] is approximately [tex]\(1.5874010519682\)[/tex], and simplifying gives:
[tex]\[
\frac{1.5874010519682 x^{4/3}}{3 e^x}
\][/tex]
Step 9: Combine the constants.
[tex]\[
\frac{1.5874010519682}{3} \times \frac{x^{4/3}}{e^x}
\][/tex]
[tex]\[
\frac{1.5874010519682}{3} \approx 0.5291336839894
\][/tex]
So, we can write the final expression as:
[tex]\[
0.5291336839894 \cdot \frac{x^{4/3}}{e^x}
\][/tex]
Step 10: Simplify.
[tex]\[
0.5291336839894 \cdot x^{4/3} \cdot e^{-x}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
0.5291336839894 \cdot x^{4/3} \cdot e^{-x}
\][/tex]
