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Let's go over the problems one by one to simplify each expression and rewrite the result without zero and negative exponents.

### 1. Simplifying the first expression:

\[
\frac{16x^{-4}y^{\frac{1}{2}}}{27x^{3}y^{3}}
\]

Step-by-step:
- Combine the exponents for \(x\): \(x^{-4} / x^3 = x^{-4-3} = x^{-7}\).
- Combine the exponents for \(y\): \(y^{\frac{1}{2}} / y^3 = y^{\frac{1}{2} - 3} = y^{\frac{1}{2} - \frac{6}{2}} = y^{-\frac{5}{2}}\).
- The expression becomes:
  \[
  \frac{16}{27} x^{-7} y^{-\frac{5}{2}}
  \]
- Rewrite it with positive exponents:
  \[
  \frac{16}{27} \frac{1}{x^7 y^{\frac{5}{2}}}
  \]
- Alternatively, you can express \(y^{\frac{5}{2}}\) as a square root:
  \[
  \frac{16}{27} \cdot \frac{1}{x^7 \sqrt{y^5}}
  \]

### 2. Simplifying the second expression:

\[
\left( \frac{16x^{-4}y^{\frac{1}{2}}}{9x^{-2}y^{\frac{3}{2}}} \right)^{\frac{1}{2}}
\]

Step-by-step:
- Simplify inside the parentheses before applying the exponent:
  - Combine \(x\)-terms: \(x^{-4} / x^{-2} = x^{-4 + 2} = x^{-2}\).
  - Combine \(y\)-terms: \(y^{\frac{1}{2}} / y^{\frac{3}{2}} = y^{\frac{1}{2} - \frac{3}{2}} = y^{-1}\).
  - Inside the parentheses, we have:
    \[
    \frac{16}{9} x^{-2} y^{-1}
    \]
- Now apply the \(\frac{1}{2}\) exponent:
  - \(\left(\frac{16}{9}\right)^{\frac{1}{2}} = \frac{4}{3}\),
  - \(x^{-2} \cdot \frac{1}{2} = x^{-1}\),
  - \(y^{-1} \cdot \frac{1}{2} = y^{-\frac{1}{2}}\).
- The result is:
  \[
  \frac{4}{3} x^{-1} y^{-\frac{1}{2}} = \frac{4}{3} \frac{1}{x \sqrt{y}}
  \]

### 3. Simplifying the third expression:

\[
\sqrt{\frac{28x^3 y^6}{9z^2}}
\]

Step-by-step:
- Apply the square root to each part:
  \[
  \sqrt{\frac{28x^3 y^6}{9z^2}} = \frac{\sqrt{28x^3 y^6}}{\sqrt{9z^2}}
  \]
- The denominator simplifies to:
  \[
  \sqrt{9z^2} = 3z
  \]
- The numerator simplifies to:
  \[
  \sqrt{28x^3 y^6} = \sqrt{28} \cdot \sqrt{x^3} \cdot \sqrt{y^6}
  \]
  - \(\sqrt{28} = 2\sqrt{7}\),
  - \(\sqrt{x^3} = x^{\frac{3}{2}}\),
  - \(\sqrt{y^6} = y^3\).
- The result is:
  \[
  \frac{2\sqrt{7}x^{\frac{3}{2}} y^3}{3z}
  \]

### 4. Simplifying the fourth expression:

\[
\sqrt[3]{375m^3n^4}
\]

Step-by-step:
- Apply the cube root:
  \[
  \sqrt[3]{375m^3n^4} = \sqrt[3]{375} \cdot \sqrt[3]{m^3} \cdot \sqrt[3]{n^4}
  \]
  - \(\sqrt[3]{375} = 5 \sqrt[3]{15}\),
  - \(\sqrt[3]{m^3} = m\),
  - \(\sqrt[3]{n^4} = n \cdot \sqrt[3]{n}\).
- The result is:
  \[
  5m n \sqrt[3]{15n}
  \]

### 5. Simplifying the fifth expression:

\[
\sqrt[3]{\frac{a^6b^{15}}{a^8b^{5}}}
\]

Step-by-step:
- Simplify inside the cube root:
  \[
  \frac{a^6b^{15}}{a^8b^5} = a^{6-8}b^{15-5} = a^{-2}b^{10}
  \]
- Now apply the cube root:
  \[
  \sqrt[3]{a^{-2}b^{10}} = a^{-\frac{2}{3}} b^{\frac{10}{3}}
  \]
- Rewrite it with positive exponents:
  \[
  \frac{b^{\frac{10}{3}}}{a^{\frac{2}{3}}}
  \]

---

Would you like more details on any specific step or explanation?

Here are five related questions:
1. How do you combine exponents when multiplying or dividing terms with the same base?
2. How does taking a square root or cube root affect exponents?
3. Can you explain how to simplify negative exponents in expressions?
4. What is the process for simplifying a fraction raised to an exponent?
5. How do you rewrite expressions without fractional exponents?

**Tip**: When simplifying expressions with exponents, always reduce the powers first before applying any root or fractional powers to make the process easier.

Simplify in lowest terms and write the result without zero and negative exponent.

Step-by-step guide on simplifying algebraic expressions with exponents and radicals. This exercise includes handling negative exponents, fractional powers, and square roots, suitable for students in Grades 9-11.

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