The nth Root and Fractional Indices

The nth root of a number is the value that, when raised to the power of $n$, gives the original number. It is written as:

$$\sqrt[n]{x}$$

Examples:

• Square root ($2^{nd}$ root):
  $$\sqrt{9}=3\text{(Since }{3}^2=9\text{)}$$
• Cube root ($3^{rd}$ root):
  $$\sqrt[3]{125}=5\text{(Since }{5}^3=125\text{)}$$
• Fourth root ($4^{th}$ root):
  $$\sqrt[4]{16}=2\text{(Since }{2}^4=16\text{)}$$

The nth root is useful in algebra, geometry, and real-world applications like physics and finance.

Mathematically, the nth root follows this rule:

$$\sqrt[n]{a}\times \sqrt[n]{a}\times \cdots \times \sqrt[n]{a}=a$$

where the nth root appears n times in multiplication.

• Example 1: If you multiply $\sqrt[3]{8}$ three times, you get 8.

$$\sqrt[3]{8}\times \sqrt[3]{8}\times \sqrt[3]{8}=8$$

• Example 2: If you multiply $\sqrt[4]{16}$ four times, you get 16.

$$\sqrt[4]{16}\times \sqrt[4]{16}\times \sqrt[4]{16}\times \sqrt[4]{16}=16$$

The Fractional Exponent Rule is a fundamental rule in algebra that connects exponents and radicals (roots): 

$$a^{\frac{1}{n}}=\sqrt[n]{a}$$

This means, instead of writing the nth root using the radical symbol $\sqrt[n]{a}$, we can express it as an exponent $a^{\frac{1}{n}}$.

Examples:

$${125}^{\frac{1}{3}}=\sqrt[3]{125}=5$$

The fractional index rule states that an exponent in the form of a fraction can be rewritten using roots:

$$a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m$$

This means:

• The denominator ($n$) represents the nth root.
• The numerator ($m$) represents the power applied after taking the root.

For example:

1. Rewrite the fractional exponent:
   $${125}^{\frac{2}{3}}={125}^{\frac{1}{3}\times 2}={\left({125}^{\frac{1}{3}}\right)}^2$$
2. Find the cube root of 125:
   $${125}^{\frac{1}{3}}=\sqrt[3]{125}=5$$
3. Square the result:
   $$5^2=25$$
4. Thus:
   $${125}^{\frac{2}{3}}=25$$

By following this fractional index rule, you can simplify fractional exponents.