What is an Exponent?
An exponent is a mathematical operation that describes how many times a number, called the base, is multiplied by itself. It is often represented as a small number written above and to the right of the base. This small number is called the exponent or power.
Basic Concept:
For a number a raised to the power of n, denoted as an, it means multiplying
a by itself n times. For example:
23 = 2 × 2 × 2 = 8
Here, 2 is the base, and 3 is the exponent, meaning 2 is multiplied by itself three times to get 8.
Important Exponent Rules:
- Product of Powers Rule: When multiplying two powers with the same base, add the exponents:
am × an = a(m+n) - Power of a Power Rule: When raising a power to another power, multiply the exponents:
(am)n = a(m × n) - Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents:
am ÷ an = a(m-n) - Power of a Product Rule: When raising a product to a power, apply the exponent to each factor:
(a × b)n = an × bn - Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1:
a0 = 1 (for a ≠ 0) - Negative Exponent Rule: A negative exponent means taking the reciprocal of the base raised to the positive of that exponent:
a(-n) = 1 / an - Fractional Exponent Rule: A fractional exponent can be rewritten as a root:
a(m/n) = √[n]am
Example Problems:
34 = 3 × 3 × 3 × 3 = 815(-2) = 1 / 52 = 1 / 2523 × 22 = 2(3+2) = 25 = 32(42)3 = 4(2 × 3) = 46 = 4096
Real-World Applications:
Exponents appear in many fields such as:
- Scientific Notation: Exponents are used to express very large or small numbers. For example,
6.022 × 1023(Avogadro’s number). - Computer Science: Exponents are used in algorithms and in expressing time complexity.
- Physics: In exponential growth and decay, like in population models, radioactive decay, or compound interest.
Understanding exponents is crucial for higher-level mathematics, algebra, calculus, and many applied science disciplines.