Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. This concept is fundamental in many areas of mathematics, including algebra, calculus, and more.

Definition

An exponent refers to the number of times a number is multiplied by itself. For example, 2^3 means 2 is multiplied by itself 3 times: 2 * 2 * 2 = 8.

In this example, 2 is the base, and 3 is the exponent or power. The whole expression 2^3 is called a power.

Properties

Here are some important properties of exponents:

1. Product of Powers: a^n * a^m = a^(n+m). This property states that when you multiply two powers with the same base, you can add the exponents.
2. Quotient of Powers: a^n / a^m = a^(n-m). This property states that when you divide two powers with the same base, you can subtract the exponents.
3. Power of a Power: (a^n)^m = a^(n*m). This property states that an exponent raised to another exponent means you multiply the exponents.
4. Zero Exponent: a^0 = 1. This property states that any non-zero number raised to the power of zero equals 1.
5. Negative Exponent: a^(-n) = 1 / a^n. This property states that a negative exponent means the reciprocal of the base raised to the opposite positive power.

Applications

Exponents are used widely in mathematics to solve a variety of problems. They are fundamental in fields such as algebra, calculus, and physics. For example, they are used to express large quantities, decay rates, population growth, compound interest, and many other natural and social phenomena.