To solve expressions with fractional indices (exponents), you follow a method that connects the fractional exponent with roots and powers. Here's the general approach:

### General Formula:
\[
a^{\frac{m}{n}} = \left( \sqrt[n]{a} \right)^m = \left( a^m \right)^{\frac{1}{n}}
\]
Where:
- \( a \) is the base,
- \( m \) is the numerator of the fraction (which represents the power),
- \( n \) is the denominator (which represents the root).

### Steps to Solve:
1. **Rewrite the fractional exponent** using the rule: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \) or \( a^{\frac{m}{n}} = \left( \sqrt[n]{a} \right)^m \).
   - The denominator \( n \) corresponds to the root of the number \( a \).
   - The numerator \( m \) corresponds to the power of the number \( a \).

2. **Simplify the root**: Take the \( n \)-th root of the base first.
3. **Apply the power**: Once you’ve found the root, raise it to the power \( m \).

#### Example 1: Solve \( 16^{\frac{3}{2}} \)
- Step 1: Rewrite the expression. 
  \[
  16^{\frac{3}{2}} = \left( \sqrt{16} \right)^3
  \]
- Step 2: Find the square root of 16 (since the denominator is 2).
  \[
  \sqrt{16} = 4
  \]
- Step 3: Raise the result to the power of 3 (since the numerator is 3).
  \[
  4^3 = 64
  \]
So, \( 16^{\frac{3}{2}} = 64 \).

#### Example 2: Solve \( 27^{\frac{2}{3}} \)
- Step 1: Rewrite the expression.
  \[
  27^{\frac{2}{3}} = \left( \sqrt[3]{27} \right)^2
  \]
- Step 2: Find the cube root of 27 (since the denominator is 3).
  \[
  \sqrt[3]{27} = 3
  \]
- Step 3: Raise the result to the power of 2 (since the numerator is 2).
  \[
  3^2 = 9
  \]
So, \( 27^{\frac{2}{3}} = 9 \).

### Summary:
- **Fractional indices** are essentially combinations of roots and powers.
- The **denominator** represents the root and the **numerator** represents the power.

Would you like to go over more details or examples? Here are some related questions to explore:

1. How do you simplify \( x^{\frac{m}{n}} \) when \( x \) is negative?
2. How do you solve equations with fractional indices?
3. Can fractional indices result in irrational numbers?
4. What happens when the fractional index has a negative exponent?
5. How do you apply fractional indices in real-world problems?

**Tip:** Always simplify the root first before applying the power—it makes calculations easier.

Learn how to solve expressions with fractional indices, connecting fractional exponents to roots and powers. This step-by-step guide covers examples such as 16^(3/2) and 27^(2/3), suitable for students in Grades 9-11.

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