# Exponents: Laws and Properties

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m+n

m-n

mn

1000

three

a^n / b^n

positive

fractional

independent

decay

## Exponents: Unlocking Powers of Numbers

Exponents, also known as indices or powers, are a fundamental concept in mathematics that allow us to express and manipulate numbers in specific ways. Let's dive into the world of exponents, beginning with the laws that govern this system.

### Definition of an Exponent

An exponent, denoted by the superscript that comes after a number or variable, represents the number of times the base is multiplied by itself. For example, in the expression (4^3), the exponent 3 indicates that 4 is multiplied by itself three times, resulting in (4 \times 4 \times 4 = 64).

### Laws of Exponents

1. Multiplication property of exponents: If (a) is a number and (m,n) are positive integers, then (a^m \times a^n = a^{m+n}). For instance, (2^3 \times 2^2 = 2^5 = 32).

2. Division property of exponents: If (a) is a nonzero number and (m,n) are positive integers, then (a^m \div a^n = a^{m-n}). For example, (\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27).

3. Power of a power property: If (a) is a nonzero number and (m,n) are integers, then ((a^m)^n = a^{mn}). For instance, ((4^2)^3 = 4^{2 \times 3} = 4^6 = 4096).

4. Product to a power: If (a) and (b) are nonzero numbers and (n) is an integer, then ((ab)^n = a^n \times b^n). For example, ((2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000).

5. Quotient to a power: If (a) and (b) are nonzero numbers and (n) is an integer, then (\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}). For instance, (\left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27}).

These laws provide a framework for manipulating exponents and simplifying expressions.

### Negative Exponents

Negative exponents are used to represent the reciprocal of a base raised to a positive exponent. For example, (a^{-n}) is the same as (\frac{1}{a^n}).

### Fractional Exponents

Fractional exponents, also known as roots, represent the result of raising the base to a fractional power. For instance, (a^{\frac{1}{n}}) is the same as the (n^{th}) root of (a).

### Exponential Functions

Exponents are closely related to exponential functions, which typically take the form (y = a^x), where (a) is the base and (x) is the independent variable. The behavior of exponential functions is characterized by rapid growth (when (a > 1)) and decay (when (0 < a < 1)).

Exponents and the laws governing them are fundamental tools in mathematics, allowing us to simplify and analyze various expressions and functions. By mastering the laws of exponents, you will lay a firm foundation in algebra, geometry, and calculus.

Dive into the world of exponents, exploring the definition and laws that govern these powerful mathematical tools. Learn about multiplication, division, power of a power, product to a power, and quotient to a power properties of exponents. Understand how negative and fractional exponents, as well as exponential functions, play a crucial role in algebra and beyond.

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