Properties of Exponents

Properties of Exponents

Glossary

  • Base: The number being multiplied.

  • Power: The exponent itself, indicating how many times the base is used as a factor.

  • Exponent: Indicates the number of times the base is multiplied.

  • Simplify: To reduce to a less complicated form by dividing common factors, regrouping similar terms, and ensuring fractions are in simplest form with no negative or zero exponents.

  • Monomials: Algebraic expressions consisting of one term.

Objective

  • Understand and apply the laws of exponents.

Exponential Expressions

An exponential expression is written as b^n, where:

  • b is the base (factor repeatedly multiplied).

  • n is the exponent (how many times b is multiplied).

Example:

  • 4^5 = 4 × 4 × 4 × 4 × 4

  • x^3 = x × x × x

Negative Exponents

An expression with a negative power is the reciprocal of the same expression with a positive exponent:

  • b^(-n) = 1 / b^n (when b ≠ 0).

Example:

  • 4^(-5) = 1 / 4^5

  • y^(-3) = 1 / y^3 (for y ≠ 0).

Zero Exponents

The value of any non-zero base raised to the power of zero is always 1:

  • b^0 = 1 (for b ≠ 0).

Example:

  • 4^0 = 1

  • x^0 = 1 (for x ≠ 0).

Laws of Exponents

Definitions apply to real numbers a, b and integers m, n.


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


  • Power of a Power:(a^m)^n = a^(mn)


  • Power of a Product:(ab)^m = a^m × b^m


  • Quotient of Powers:a^m / a^n = a^(m-n) (for a ≠ 0).


  • Power of a Quotient:(a / b)^m = a^m / b^m (for b ≠ 0).

Simplifying Expressions

Use laws of exponents to simplify monomials. Monomials: expressions with just one term, having no negative exponents and only one of each variable.

Example of simplified form: 5ab^2c^3 is correct, but (3x^2y^(-2))^3 is not simplified due to parentheses and negative exponent.

Example of Simplifying Expressions

Expression: (3x^2y^(-2))^3

  1. Start by eliminating the parentheses: (3x^2y^(-2))^3 = 3^3 × (x^2)^3 × (y^(-2))^3 = 27x^(6)y^(-6)

  2. Writing with Positive Exponents: y^(-6) = 1 / y^6, resulting in 27x^6y^6.

Second Example: (-5cd^(-4))(2cd^2)^2

  • Raise factors in the parentheses to the second power: -5cd^(-4) × (2^2c^2d^(4)) = -5c^(1)d^(-4) × 4cd^2.

  • Multiply powers with the same base by adding the exponents: Final result: -20c^(3)d^(-2) can be expressed using positive exponents.

Ratios of Monomial Expressions

Use laws of exponents to simplify ratios of two monomials. Ensure no negative exponents, one of each variable, and no parentheses for full simplification.

Conclusion

The laws of exponents are essential for simplifying expressions and understanding negative exponents in relation to their reciprocal forms.