The nth Root and Fractional Indices

🔮 Definition of nth Root:

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:

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

🔮 Examples of nth Root:

• 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):
  $$ \sqrt[4]{16} = 2 \quad \text{(Since } 2^4 = 16 \text{)} $$

🔮 What is the nth Root Rule?

The nth root rules tells us, if you multiply the nth root of a number exactly n times, you get back the original number.

$$
\Large \underbrace{\sqrt[n]{a} \times \sqrt[n]{a} \times \cdots \times \sqrt[n]{a}}_{\text{n times}} = a
$$

• 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.

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

🔮 What Is the Fractional Exponent Rule?

The Fractional Exponent Rule connects exponents and roots:

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

It tells us that taking the nth root of a number is the same as raising it to the power of $ \frac{1}{n}$.

🔍 Example of Using This Rule:

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

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

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

This means:

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

🔍 Example of Simplifying Fractional Index:

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:
   $$ 125^{\frac{2}{3}} = 25 $$