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

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

Mathematically, the nth root follows this rule:

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

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.