Exponents: Foundations, Properties, and Advanced Concepts

Exponents are a fundamental concept in mathematics, representing repeated multiplication. They appear across all levels of math, from arithmetic and algebra to calculus, number theory, and complex analysis. In this write-up, we will explore their definitions, properties, generalizations, and applications.


1. Definition of Exponents

For any real number and positive integer , the expression (read as “ raised to the th power”) is defined as:


This is known as exponential notation. For example:


2. Special Cases and Extension

We extend the definition of exponents beyond positive integers:

• Zero exponent: for all
• Negative exponents:
• Fractional exponents: , and more generally
• Rational exponents are defined via roots and powers, and extend to real and even complex exponents using logarithms and limits

Example:


3. Properties of Exponents

For all real (or complex) numbers and integers (or rationals, where defined) :

• 
• , assuming
• 
• 
• , assuming

These identities are essential in simplifying expressions and solving exponential equations.

4. Exponents and the Order of Operations

Exponents are evaluated before multiplication and addition in expressions. For example:


5. Exponential Growth and Decay

Exponents model natural phenomena such as population growth, radioactive decay, and compound interest. A typical exponential growth formula is:


where:

• is the population at time
• is the initial population
• is the growth rate
• is Euler’s number

Exponential decay is modeled similarly with .

6. Exponential Functions

The exponential function is defined as:


Special case: , which is especially important in calculus due to its unique property:


7. Exponents in Modular Arithmetic

Exponents also appear in modular arithmetic. For example, Euler’s theorem states:

If , then:


This is a generalization of Fermat’s Little Theorem:


when is prime and .

8. Exponential Identities with Logarithms

Exponents are inverse operations to logarithms. For , :

• 
• 
• 
• Change-of-base formula:

9. Exponentiation in Complex Numbers

Using Euler’s formula, complex exponentials can be written as:


This leads to the identity:


which elegantly connects the fundamental constants , , and .

10. Generalizations: Power Series and Binomial Expansions

Using exponents, we define power series such as:


And more generally, using the Binomial Theorem for arbitrary exponents :


where


11. Conclusion

Exponents form the foundation of much of mathematics and science. From simple powers to complex exponentials and exponential functions, they offer a powerful language for expressing and modeling mathematical relationships.

References

• Wikipedia: Exponentiation
• AoPS Wiki: Exponent
• Stewart, J. Calculus, Cengage Learning
• Spivak, M. Calculus, Cambridge Press