Exponentiation

Exponentiation is a mathematical operation that represents repeated multiplication of the same number. The number being multiplied is called the base, and the number of times it is multiplied by itself is called the exponent. The general format for exponentiation is:

base^exponent

For example, 2³ represents 2 multiplied by itself 3 times, which equals 8 (2 x 2 x 2).

Properties of Exponents

1. Product of Powers: When multiplying two powers with the same base, add the exponents.
   • a^m * aⁿ = a^m+n
   • Example: 2³ * 2⁴ = 2⁷
2. Quotient of Powers: When dividing two powers with the same base, subtract the exponents.
   • a^m / aⁿ = a^m-n
   • Example: 3⁵ / 3² = 3³
3. Power of a Power: When raising a power to another exponent, multiply the exponents.
   • (a^m)ⁿ = a^{m * n}
   • Example: (4³)² = 4⁶
4. Power of a Product: When raising a product to an exponent, distribute the exponent to each factor.
   • (ab)ⁿ = aⁿ * bⁿ
   • Example: (5 * 6)² = 5² * 6²

Practice Problems

1. Calculate 2⁴.
2. Simplify: 3² * 3⁵.
3. What is the result of (7³)²?
4. Simplify: (4 * 9)².

By mastering these properties and practicing the problems, you'll have a solid understanding of exponentiation. Good luck!