Understanding Exponents

• An exponent is a number that indicates how many times a base is multiplied by itself.
  • For example, in 2³, the base is 2 and the exponent is 3. This means 2 * 2 * 2.
• The result of raising a base to an exponent is called a power.

Laws of Exponents

1. Product Rule: When multiplying powers with the same base, add the exponents.
   • a^m * a^n = a^(m+n)
   • Example: 2³ * 2⁴ = 2^(3+4) = 2⁷
2. Quotient Rule: When dividing powers with the same base, subtract the exponents.
   • a^m / a^n = a^(m-n)
   • Example: 3⁵ / 3² = 3^(5-2) = 3³
3. Power Rule: When raising a power to another power, multiply the exponents.
   • (a^m)^n = a^(m*n)
   • Example: (5²)³ = 5^(2*3) = 5⁶
4. Zero Exponent Rule: Any nonzero number raised to the power of 0 is 1.
   • a^0 = 1 (where a ≠ 0)
   • Example: 7⁰ = 1
5. Negative Exponent Rule: A nonzero number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent.
   • a^(-n) = 1/a^n (where a ≠ 0)  
   • Example: 2^(-3) = 1/2³ = 1/8

Applications of Exponents

• Scientific notation: A way to express very large or very small numbers.
• Compound interest: Calculating the growth of money over time.
• Population growth and decay: Modeling changes in population size.
• Exponential functions: Used in various fields, including biology, economics, and physics.

Learn with an example