Understanding Exponents: A Comprehensive Guide

What is Exponentiation?
Exponentiation is a mathematical operation expressed as ( a^n ), where ( a ) (the base) is multiplied by itself repeatedly, and ( n ) (the exponent) indicates how many times this multiplication occurs. For positive integers ( n ):
[ a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ times}} ]
For example, ( 2^3 = 2 \times 2 \times 2 = 8 ).

Calculator Notes

• Accepts negative bases (e.g., (-2^3 = -8)) but cannot compute imaginary numbers.
• Does not support fractional bases directly, but fractional exponents (e.g., ( 4^{1.5} )) can be calculated using decimal inputs.

Core Exponent Rules

1. Multiplying Same-Base Exponents
   Add the exponents:
   [ a^n \times a^m = a^{n + m} ]
   Example:
   ( 2^2 \times 2^4 = 4 \times 16 = 64 ) or ( 2^{2+4} = 2^6 = 64 ).
2. Negative Exponents
   Reciprocate the base and apply the positive exponent:
   [ a^{-n} = \frac{1}{a^n} ]
   Example:
   ( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} ).
3. Dividing Same-Base Exponents
   Subtract the exponents:
   [ \frac{a^m}{a^n} = a^{m – n} ]
   Example:
   ( \frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4 ).
4. Exponent Raised to Another Exponent
   Multiply the exponents:
   [ (a^m)^n = a^{m \times n} ]
   Example:
   ( (2^2)^4 = 2^{2 \times 4} = 2^8 = 256 ).
5. Distributing Over Multiplication/Division
   Apply the exponent to each factor:
   [ (a \times b)^n = a^n \times b^n \quad \text{and} \quad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} ]
   Examples:
   • ( (2 \times 4)^2 = 2^2 \times 4^2 = 4 \times 16 = 64 ).
   • ( \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} ).

Special Cases

• Exponent of 1:
  ( a^1 = a ).
• Exponent of 0:
  ( a^0 = 1 ) for any ( a \neq 0 ). (Note: ( 0^0 ) is debated but often defined as 1 for practicality.)
  Reasoning: If ( a^n \times a^0 = a^{n+0} = a^n ), then ( a^0 ) must equal 1 to preserve the equality.
• Fractional Exponents:
  A numerator ( m ) in ( a^{m/n} ) means:
  [ a^{m/n} = \left(\sqrt[n]{a}\right)^m ]
  Example: ( 8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4 ).

Negative Bases

• Even Exponent: Result is positive (e.g., ((-2)^4 = 16)).
• Odd Exponent: Result is negative (e.g., ((-2)^3 = -8)).
• Fractional Exponents: May yield imaginary numbers (e.g., ((-4)^{1/2} = 2i)), which this calculator cannot compute (returns “NAN”).

Final Notes
Exponent rules simplify complex calculations, but always verify input constraints (e.g., avoid negative bases with fractional exponents unless working with imaginary numbers). Use decimal forms for fractional exponents in calculations.