Exponents with negative bases

Learning Objectives Identify the base and exponent in expressions involving negative numbers. Distinguish between expressions like `(-a)^n` and `-a^n`. Evaluate exponential expressions with negative bases and even exponents. Evaluate exponential expressions with negative bases and odd exponents. Apply the order of operations correctly when evaluating expressions with negative bases. Explain why the sign of the result changes based on whether the exponent is even or odd. Solve problems involving negative bases in various mathematical contexts. Have you ever wondered what happens when you multiply a negative number by itself many times? 🤔 Sometimes the answer is positive, and sometimes it's negative! In this lesson, we'll explore the fascinating world of exponent...

TermDefinitionExample BaseThe number that is multiplied by itself in an exponential expression.In `(-3)^4`, the base is -3. ExponentThe small number written above and to the right of the base, indicating how many times the base is used as a factor.In `(-3)^4`, the exponent is 4. PowerThe entire expression consisting of a base and an exponent.`(-3)^4` is a power. Negative BaseWhen the number being multiplied by itself is a negative number. It is usually enclosed in parentheses.`(-5)^2` has a negative base of -5. ParenthesesSymbols `()` used to group numbers and operations, clearly indicating what part of an expression is the base.In `(-2)^3`, the parentheses show that -2 is the base. In `-2^3`, only 2 is the base. Even ExponentAn exponent that is an even number (2, 4, 6, ...).In `(-7)^2`,...

Negative Base with Even Exponent $$(-a)^n = a^n \text{ (if n is even)}$$ When a negative base is raised to an even exponent, the result is always positive. This is because an even number of negative factors will cancel out to positive pairs. Negative Base with Odd Exponent $$(-a)^n = -(a^n) \text{ (if n is odd)}$$ When a negative base is raised to an odd exponent, the result is always negative. This is because an odd number of negative factors will result in a negative product. Negative Sign Outside Parentheses $$-a^n = -(a^n)$$ If there are no parentheses around the negative base, the negative sign is NOT part of the base. The exponent only applies to the number immediately next to it. The negative sign is applied AFTER the exponentiation.