Definition and Basic Rules of Exponents
Exponents represent repeated multiplication of a base number. For instance, a^n means a multiplied by itself n times. The fundamental rules include: the product rule (a^m × a^n = a^{m+n}), the quotient rule (a^m ÷ a^n = a^{m-n}), the power rule ((a^m)^n = a^{m×n}), the zero exponent rule (a^0 = 1 for a ≠ 0), and the negative exponent rule (a^{-n} = 1 / a^n).
Key Principles and Additional Rules
Other important principles are the product-to-a-power rule ( (a × b)^n = a^n × b^n ) and the quotient-to-a-power rule ( (a / b)^n = a^n / b^n ). These rules apply when bases are the same for product and quotient operations. For different bases, multiplication and division occur separately without combining exponents.
Practical Examples
Consider simplifying 2^3 × 2^4: using the product rule, this equals 2^{3+4} = 2^7 = 128. For (3^2)^3, the power rule gives 3^{2×3} = 3^6 = 729. A negative exponent example: 5^{-2} = 1 / 5^2 = 1/25, useful in fractions and scientific notation.
Importance and Real-World Applications
Exponent rules simplify complex expressions in algebra, calculus, and physics, such as calculating growth rates (e.g., population models using 2^n) or electrical engineering (e.g., power laws in circuits). They enable efficient computation in scientific notation for very large or small numbers, like distances in astronomy.