Exponent Laws

Exponent laws are essential tools in Pre-Algebra that simplify calculations involving powers. Understanding how to manipulate exponents—whether multiplying, dividing, or dealing with zero and negative values—makes working with expressions much easier and more efficient.

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

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

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.
   • Example: a^5 ÷ a^2 = a^(5-2) = a^3.
   • This law helps in simplifying fractions with exponents.

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

   • When raising a power to another power, multiply the exponents.
   • Example: (a^2)^3 = a^(2×3) = a^6.
   • This law is useful for simplifying expressions with nested exponents.

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.
   • Example: (2x)^3 = 2^3 × x^3 = 8x^3.
   • This law allows for easier manipulation of products raised to powers.

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.
   • Example: (x/2)^4 = x^4 / 2^4 = x^4 / 16.
   • This law simplifies expressions involving fractions raised to powers.

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

   • Any non-zero base raised to the power of zero equals one.
   • Example: 5^0 = 1.
   • This law establishes a consistent value for zero exponents.

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

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

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

   • A fractional exponent represents a root; the denominator indicates the root and the numerator indicates the power.
   • Example: x^(1/2) = √x.
   • This law connects exponents with roots, facilitating calculations involving both.