Answer:The laws of exponents, also known as the rules of exponents, are a set of mathematical principles that govern the manipulation and simplification of expressions involving exponential notation (numbers raised to a power). These laws are essential for working with exponential functions and algebraic expressions. Here are the fundamental laws of exponents:
Product Rule: When you multiply two exponential expressions with the same base, you can add their exponents. In other words, for any positive real number 'a' and integers 'm' and 'n':
a^m * a^n = a^(m + n)
For example, 2^3 * 2^4 = 2^(3 + 4) = 2^7.
Quotient Rule: When you divide two exponential expressions with the same base, you can subtract the exponent in the denominator from the exponent in the numerator. In other words, for any positive real number 'a' and integers 'm' and 'n':
(a^m) / (a^n) = a^(m - n)
For example, (3^5) / (3^2) = 3^(5 - 2) = 3^3.
Power Rule: When you raise an exponential expression to another exponent, you can multiply the exponents. In other words, for any positive real number 'a' and integers 'm' and 'n':
(a^m)^n = a^(m * n)
For example, (4^3)^2 = 4^(3 * 2) = 4^6.
Negative Exponent Rule: A negative exponent indicates taking the reciprocal of the base raised to the positive exponent. In other words, for any positive real number 'a' and a positive integer 'n':
a^(-n) = 1 / (a^n)
For example, 2^(-3) = 1 / (2^3) = 1/8.
Zero Exponent Rule: Any nonzero number raised to the power of zero is equal to 1. In other words, for any nonzero real number 'a':
a^0 = 1
For example, 5^0 = 1.
Fractional Exponent Rule: A fractional exponent represents taking the nth root of a number. In other words, for any positive real number 'a' and a positive integer 'n':
a^(1/n) = √(n√a)
For example, 8^(1/3) represents the cube root of 8, which is 2 because 2^3 = 8.
These laws of exponents are fundamental in algebra and calculus and are used extensively when working with exponential functions and expressions. They allow for the simplification and manipulation of complex mathematical expressions involving exponents.
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