Properties of exponents

Learning Objectives Simplify expressions by applying the product and quotient of powers properties. Simplify expressions using the power of a power, power of a product, and power of a quotient properties. Correctly interpret and apply the rules for zero and negative exponents. Simplify complex algebraic expressions that require combining multiple exponent properties. Evaluate numerical expressions involving integer exponents. Apply the properties of exponents to solve problems involving scientific notation. How can we describe the distance to the nearest star, over 40,000,000,000,000 km away, without writing all those zeros? 🔭 Exponents are a powerful shorthand for repeated multiplication. Mastering their properties is a foundational skill in algebra, allowing you to simpl...

TermDefinitionExample BaseThe number or variable that is being multiplied by itself.In the expression 5³, the base is 5. Exponent (or Power)A number that indicates how many times the base is used as a factor in a multiplication.In the expression 5³, the exponent is 3. This means 5 * 5 * 5. PowerThe entire expression consisting of a base and an exponent.The expression 5³ is a power. Zero ExponentAny non-zero base raised to the power of zero is equal to 1.x⁰ = 1 (for x ≠ 0), and 1,257⁰ = 1. Negative ExponentAn exponent that indicates the reciprocal of the base raised to the corresponding positive exponent.x⁻ⁿ = 1/xⁿ. For instance, 3⁻² = 1/3² = 1/9.

Product of Powers xᵃ * xᵇ = xᵃ⁺ᵇ When multiplying powers with the same base, keep the base and add the exponents. Quotient of Powers xᵃ / xᵇ = xᵃ⁻ᵇ When dividing powers with the same base, keep the base and subtract the exponents (numerator exponent minus denominator exponent). Power of a Power (xᵃ)ᵇ = xᵃ*ᵇ When raising a power to another power, keep the base and multiply the exponents. Power of a Product & Quotient (xy)ᵃ = xᵃyᵃ and (x/y)ᵃ = xᵃ/yᵃ When a product or quotient is raised to a power, distribute the power to each factor in the base.