Exponents Rules

Understanding exponent rules is key in Algebra and Trigonometry, Elementary Algebra. These rules simplify calculations involving powers, making it easier to work with expressions. Mastering them will help you tackle more complex problems with confidence.

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

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

2. Quotient Rule: 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 rule helps simplify expressions involving division of like bases.
   • Example: 5^4 / 5^2 = 5^(4-2) = 5^2 = 25.

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

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

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

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

5. Power of a Quotient Rule: (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 rule simplifies calculations involving fractions raised to powers.
   • Example: (4/2)^2 = 4^2 / 2^2 = 16 / 4 = 4.

6. Zero Exponent Rule: a^0 = 1 (where a ≠ 0)

   • Any non-zero base raised to the power of zero equals one.
   • This rule is essential for understanding the behavior of exponents in various expressions.
   • Example: 7^0 = 1.

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

   • A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.
   • This rule helps convert negative exponents into a more manageable form.
   • Example: 2^(-3) = 1 / 2^3 = 1 / 8.

8. Fractional Exponent Rule: a^(m/n) = ⁿ√(a^m)

   • A fractional exponent represents both a power and a root; the numerator is the power and the denominator is the root.
   • This rule is useful for simplifying expressions involving roots and powers.
   • Example: 8^(1/3) = ⁿ√(8) = 2.