Laws of Exponents

The laws of exponents, also known as the rules of indices, are a set of mathematical principles that dictate how to manipulate and simplify expressions involving exponents. Exponents are superscripts used to represent the repeated multiplication of a number or variable. These laws provide a systematic approach to handle various operations, such as addition, subtraction, multiplication, and division, when dealing with exponential expressions. The fundamental laws of exponents are as follows: Product of Powers: When multiplying two exponential expressions with the same base, add their exponents. a^m * a^n = a^(m+n) Quotient of Powers: When dividing two exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator. a^m / a^n = a^(m-n) Power of a Power: When raising an exponential expression to another power, multiply the exponents. (a^m)^n = a^(m*n) Power of a Product: When raising a product of two numbers or variables to a power, distribute the exponent to each factor. (ab)^m = a^m * b^m Power of a Quotient: When raising a quotient of two numbers or variables to a power, distribute the exponent to both the numerator and the denominator. (a/b)^m = a^m / b^m Zero Exponent: Any non-zero number or variable raised to the power of zero is equal to one. a^0 = 1, where a ≠ 0 Negative Exponent: A number or variable raised to a negative power is equal to the reciprocal of the same number or variable raised to the corresponding positive power. a^(-m) = 1/a^m, where a ≠ 0 These laws are essential for simplifying and solving algebraic and calculus problems involving exponential expressions, allowing for efficient manipulation and evaluation of complex mathematical relationships.