Properties of Exponents

Understanding the properties of exponents is key in Intermediate Algebra. These rules simplify calculations involving powers, making it easier to work with expressions and equations. Mastering these concepts will enhance your problem-solving skills and boost your confidence in algebra.

1. Product of Powers: a^m * a^n = a^(m+n)

   • When multiplying two powers with the same base, add their exponents.
   • This property simplifies calculations involving powers.
   • Example: 2^3 * 2^2 = 2^(3+2) = 2^5 = 32.

2. Quotient of Powers: a^m / a^n = a^(m-n)

   • When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
   • This property helps in simplifying fractions with exponents.
   • Example: 5^4 / 5^2 = 5^(4-2) = 5^2 = 25.

3. Power of a Power: (a^m)^n = a^(m*n)

   • When raising a power to another power, multiply the exponents.
   • This property is useful for simplifying expressions with nested exponents.
   • Example: (3^2)^4 = 3^(2*4) = 3^8 = 6561.

4. Power of a Product: (ab)^n = a^n * b^n

   • When raising a product to a power, distribute the exponent to each factor in the product.
   • This property allows for easier manipulation of products with exponents.
   • Example: (2*3)^3 = 2^3 * 3^3 = 8 * 27 = 216.

5. Power of a Quotient: (a/b)^n = a^n / b^n

   • When raising a quotient to a power, distribute the exponent to both the numerator and the denominator.
   • This property simplifies expressions involving fractions with exponents.
   • Example: (4/2)^2 = 4^2 / 2^2 = 16 / 4 = 4.

6. Zero Exponent: a^0 = 1 (a ≠ 0)

   • Any non-zero base raised to the power of zero equals one.
   • This property is essential for understanding limits and continuity in algebra.
   • Example: 7^0 = 1.

7. Negative Exponent: a^(-n) = 1 / a^n

   • A negative exponent indicates the reciprocal of the base raised to the positive exponent.
   • This property helps in converting negative exponents into positive ones for simplification.
   • Example: 2^(-3) = 1 / 2^3 = 1 / 8.

8. Fractional Exponent: a^(m/n) = nth root of a^m

   • A fractional exponent represents both a power and a root.
   • The numerator indicates the power, while the denominator indicates the root.
   • Example: 8^(1/3) = cube root of 8 = 2.