The expression given is:

$(2 \sqrt{x})(3 \sqrt[3]{x})x > 0$

Let’s simplify this step by step, assuming $x > 0$.

Step 1: Breaking Down the Components

You have three terms in the product:

1. $2\sqrt{x}$
2. $3\sqrt[3]{x}$
3. $x$

Where:

• $\sqrt{x}$ is the square root of $x$,
• $\sqrt[3]{x}$ is the cube root of $x$.

Step 2: Multiplying Constants

The constants $2$ and $3$ can be multiplied directly:

$2 \times 3 = 6$

Step 3: Multiplying the Powers of $x$

Now, let's deal with the powers of $x$.

• $\sqrt{x}$ is equivalent to $x^{1/2}$,
• $\sqrt[3]{x}$ is equivalent to $x^{1/3}$,
• The third term is simply $x$, or $x^{1}$.

When multiplying the powers of $x$, we add the exponents:

$x^{1/2} \times x^{1/3} \times x^{1} = x^{1/2 + 1/3 + 1}$

To add the fractions $1/2$, $1/3$, and $1$, let's first get a common denominator:

$1/2 = 3/6, \quad 1/3 = 2/6, \quad 1 = 6/6$

Now, sum them:

$\frac{3}{6} + \frac{2}{6} + \frac{6}{6} = \frac{11}{6}$

Thus, the combined exponent of $x$ is $\frac{11}{6}$.

Step 4: Final Expression

Putting everything together, the simplified form of the expression is:

$6x^{11/6}$

Since $x > 0$, this is valid for positive values of $x$.

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Would you like any further details or clarifications?

Here are 5 related questions for you:

1. How would the expression change if $x$ could be negative?
2. Can you express the simplified form in logarithmic terms?
3. What happens if the exponents in the roots were different?
4. How does the process change when dealing with variables under roots that are not powers of $x$?
5. How does the behavior of the function change as $x \to \infty$?

Tip: When simplifying expressions with roots, always convert the roots into fractional exponents. This makes multiplying and combining powers much easier!