Here are some advanced questions related to indices (also called exponents or powers):

1. **Negative Exponents and Rational Exponents**  
   Simplify the expression:  
   \[
   \left( \frac{8x^6y^{-3}}{2x^{-2}y^4} \right)^{-2}
   \]

2. **Exponent Laws and Logarithmic Connection**  
   Prove that:
   \[
   \log_b(x^m) = m \cdot \log_b(x)
   \]
   Using the laws of indices, justify the relationship between exponents and logarithms.

3. **Surds and Fractional Indices**  
   Simplify and express the answer with positive exponents:  
   \[
   (4^{3/2} \times 16^{-1/2})^3
   \]

4. **Compound Powers and Mixed Indices**  
   Simplify the expression:  
   \[
   \left( 5^3 \times 25^{3/2} \right)^{4/5}
   \]

5. **Equating Expressions with Indices**  
   Solve for \(x\) in the equation:  
   \[
   3^{2x-1} \times 9^{x+2} = 27^{x+3}
   \]

### Tip:
Always simplify expressions by applying the laws of exponents first (such as \(a^m \times a^n = a^{m+n}\)) to make the equation easier to handle.

Would you like a detailed solution for any of these? Feel free to ask!

Explore advanced questions on indices, covering negative exponents, logarithmic connections, and fractional indices. Learn to simplify expressions and solve complex exponent problems, suitable for students in Grades 10-12.

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