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.