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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.

Nyt roknireglurnar fyri potensar og røtur og umskriva til ein potens: \( \sqrt[3]{11^2} \cdot \sqrt[1.5]{11^7} \)

Learn how to simplify the expression \( \sqrt[3]{11^2} \cdot \sqrt[1.5]{11^7} \) by converting roots into fractional exponents and applying exponent addition rules. This problem is suitable for students in Grades 10-12 studying advanced algebraic concepts.

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