To solve the equation [tex]\(x \sqrt[3]{x} = ( x \cdot \sqrt[3]{x })^{ x}\)[/tex], let's follow these steps:
1. Simplify the left-hand side of the equation:
[tex]\[
x \sqrt[3]{x} = x \cdot x^{1/3} = x^{1 + 1/3} = x^{4/3}
\][/tex]
2. Simplify the right-hand side of the equation.
Notice that the right-hand side, [tex]\( (x \cdot \sqrt[3]{x})^{x} \)[/tex], can be expressed as:
[tex]\[
(x \cdot \sqrt[3]{x})^{x} = (x \cdot x^{1/3})^x = (x^{1 + 1/3})^x = (x^{4/3})^x
\][/tex]
Using the property of exponents, [tex]\( (a^m)^n = a^{mn} \)[/tex], we can rewrite the above expression as:
[tex]\[
(x^{4/3})^x = x^{4x/3}
\][/tex]
3. Equate the simplified left-hand side to the simplified right-hand side:
[tex]\[
x^{4/3} = x^{4x/3}
\][/tex]
4. Since both sides have the same base, we can set their exponents equal to each other:
[tex]\[
\frac{4}{3} = \frac{4x}{3}
\][/tex]
5. Solve for [tex]\(x\)[/tex].
Multiplying both sides by 3 to eliminate the denominators:
[tex]\[
4 = 4x
\][/tex]
Dividing both sides by 4:
[tex]\[
x = 1
\][/tex]
Therefore, the solution to the equation [tex]\(x \sqrt[3]{x} = (x \cdot \sqrt[3]{x})^x\)[/tex] is:
[tex]\[
x = 1
\][/tex]
