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Let's start solving some of these expressions using properties of rational exponents. The instructions ask us to rewrite the expressions and approximate the answers to the nearest hundredth when appropriate.

### Problem 7: \((49^{\frac{1}{2}})^{\frac{3}{5}}\)
- First, rewrite the expression:
  \[
  (49^{\frac{1}{2}})^{\frac{3}{5}} = 49^{\frac{1}{2} \cdot \frac{3}{5}} = 49^{\frac{3}{10}}.
  \]
- Next, compute \(49^{\frac{3}{10}}\):
  \[
  49^{\frac{3}{10}} = (7^2)^{\frac{3}{10}} = 7^{\frac{6}{10}} = 7^{0.6}.
  \]
- Approximating \(7^{0.6}\) gives approximately **4.91**.

### Problem 8: \(\frac{4^{\frac{3}{4}}}{4^{\frac{1}{4}}}\)
- Use the quotient of powers rule: \( \frac{a^m}{a^n} = a^{m-n} \).
  \[
  \frac{4^{\frac{3}{4}}}{4^{\frac{1}{4}}} = 4^{\frac{3}{4} - \frac{1}{4}} = 4^{\frac{2}{4}} = 4^{\frac{1}{2}}.
  \]
- \(4^{\frac{1}{2}} = \sqrt{4} = 2\).

### Problem 9: \(\left(\frac{27}{125}\right)^{\frac{4}{3}}\)
- This is a power of a fraction: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).
  \[
  \left(\frac{27}{125}\right)^{\frac{4}{3}} = \frac{27^{\frac{4}{3}}}{125^{\frac{4}{3}}}.
  \]
- Calculate \(27^{\frac{4}{3}}\) and \(125^{\frac{4}{3}}\):
  \[
  27^{\frac{4}{3}} = \left(3^3\right)^{\frac{4}{3}} = 3^4 = 81,
  \]
  \[
  125^{\frac{4}{3}} = \left(5^3\right)^{\frac{4}{3}} = 5^4 = 625.
  \]
- So, \(\left(\frac{27}{125}\right)^{\frac{4}{3}} = \frac{81}{625}\), which is approximately **0.13**.

### Problem 10: \(16^7 \cdot 16^{\frac{1}{2}}\)
- Use the product of powers rule: \( a^m \cdot a^n = a^{m+n} \).
  \[
  16^7 \cdot 16^{\frac{1}{2}} = 16^{7 + \frac{1}{2}} = 16^{7.5}.
  \]
- Calculate \(16^{7.5}\):
  \[
  16^{7.5} = (2^4)^{7.5} = 2^{30}.
  \]
- \(2^{30} = 1073741824\).

### Problem 11: \((36 \cdot 64)^{\frac{3}{2}}\)
- Simplify inside the parentheses first: \(36 \cdot 64 = 2304\).
  \[
  (36 \cdot 64)^{\frac{3}{2}} = 2304^{\frac{3}{2}} = \left(\sqrt{2304}\right)^3.
  \]
- \(\sqrt{2304} = 48\), so
  \[
  48^3 = 110592.
  \]

Would you like more details on any of these steps or explanations?

Here are 5 related questions for further practice:
1. How do you simplify \(\left(\frac{64}{81}\right)^{\frac{3}{2}}\)?
2. How would you solve \((81^5 \cdot 81^{-3})\)?
3. How do you handle expressions like \(\left(49^2\right)^{\frac{1}{3}}\)?
4. What is the rule for multiplying numbers with the same base but different exponents?
5. How do you approximate \(5^{0.75}\)?

**Tip:** When working with rational exponents, always look for ways to simplify by using the properties of exponents (product, quotient, power rules).

Rewrite each expression using one of the properties of rational exponents, and then evaluate. Approximate to the nearest hundredth when appropriate.

Learn how to simplify expressions using the properties of rational exponents and evaluate the results. This example covers problems like (49^1/2)^3/5 and (27/125)^4/3, and more. Suitable for high school students.

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