The problem asks to use rules for exponents and roots to rewrite the given expression as a single power (or exponent). Let's break it down:

The expression:

[ \sqrt[3]{11^2} \cdot \sqrt[1.5]{11^7} ]

1. The cube root can be rewritten using exponents as: $\sqrt[3]{11^2} = 11^{2/3}$

2. The second term, which involves a root of $\sqrt[1.5]{11^7}$, can also be rewritten using exponents. A root of 1.5 is equivalent to a fractional power: $\sqrt[1.5]{11^7} = 11^{7/1.5} = 11^{7 \cdot \frac{2}{3}} = 11^{14/3}$

Now, multiplying the two terms:

$11^{2/3} \cdot 11^{14/3}$

When multiplying powers with the same base, you add the exponents: $11^{(2/3 + 14/3)} = 11^{16/3}$

Thus, the expression simplifies to: $11^{16/3}$

Would you like further explanation on any step? Here are five related questions you might find helpful:

1. How do you simplify products of exponents with the same base?
2. What is the general rule for converting roots to exponents?
3. How do you simplify an expression like $\sqrt[n]{x^m}$?
4. How do you handle fractional exponents in different contexts?
5. What are the rules for multiplying and dividing expressions with exponents?

Tip: Remember that roots can always be written as fractional exponents, where $\sqrt[n]{x^m} = x^{m/n}$. This makes it easier to combine powers in expressions.