The problem asks you to rewrite the expression $\sqrt{x} \cdot \sqrt[5]{x}$ in the form $c \cdot x^n$, where $c$ is a constant and $n$ is a rational number.

Let's solve this step by step:

1. Recall that $\sqrt{x} = x^{1/2}$ and $\sqrt[5]{x} = x^{1/5}$. So the expression becomes: $x^{1/2} \cdot x^{1/5}$

2. When multiplying powers of the same base, we add the exponents: $x^{1/2} \cdot x^{1/5} = x^{1/2 + 1/5}$

3. To add these fractions, find a common denominator: $1/2 + 1/5 = \frac{5}{10} + \frac{2}{10} = \frac{7}{10}$

4. So the expression becomes: $x^{7/10}$

Since there is no constant other than 1, the expression in the form $c \cdot x^n$ is:

$1 \cdot x^{7/10} \quad \text{or just} \quad x^{7/10}$

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. How would you rewrite $\sqrt[3]{x^2} \cdot \sqrt{x}$ in the form $c \cdot x^n$?
2. What is the value of $x^{1/4} \cdot x^{1/2}$?
3. How do you express $\sqrt[6]{x^3} \cdot \sqrt{x}$ in the form $x^n$?
4. If $a \cdot x^{n_1} \cdot b \cdot x^{n_2} = c \cdot x^n$, how do you determine $n$?
5. How do you express $\sqrt{x^3} \cdot \sqrt[4]{x^5}$ in exponential form?

Tip: When multiplying expressions with the same base, adding the exponents simplifies the problem quickly!