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Negative Exponents (page 2 of 5)

Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents

A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x^–2" (ecks to the minus two) just means "x², but underneath, as in 1/(x²)".

• Write x^–4 using only positive exponents.

• Write x² / x^–3 using only positive exponents.

• Write 2x^–1 using only positive exponents.

Note that the "2" above does not move with the variable; the exponent is only on the "x".

• Write (3x)^–2 using only positive exponents.

Unlike the previous exercise, the parentheses meant that the negative power did indeed apply to the three as well as the variable.

• Write (x^–2 / y^–3)^–2 using only positive exponents.

This one can also be done as:   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

Since exponents indicate multiplication, and since order doesn't matter in multiplication, there will often be more than one sequence of steps that will lead to a valid simplification of a given exercise. Don't worry if the steps in your homework look quite different from the steps in a classmate's homework. As long as your steps were correct, you should both end up with the same answer.

By the way, now that you know about negative exponents, you can understand the logic behind the "anything to the power zero" rule:

Anything to the power zero is just "1".

Why is this so? There are various explanations. One might be stated as "because that's how the rules work out." Another would be to trace through a progression like the following:

3⁵ = 3⁶ ÷ 3 = 3⁶ ÷ 3¹ = 3^6–1 = 3⁵= 243
3⁴ = 3⁵ ÷ 3 = 3⁵ ÷ 3¹ = 3^5–1 = 3⁴= 81
3³ = 3⁴ ÷ 3 = 3⁴ ÷ 3¹ = 3^4–1 = 3³= 27
3² = 3³ ÷ 3 = 3³ ÷ 3¹ = 3^3–1 = 3²= 9
3¹ = 3² ÷ 3 = 3² ÷ 3¹ = 3^2–1 = 3¹= 3

Then logically 3⁰ = 3¹ ÷ 3¹ = 3^1–1 = 3⁰ = 1.

A negative-exponents explanation of the "anything to the zero power is just 1" might be as follows:

m⁰ = m^{(n – n)} = mⁿ × m^–n = mⁿ ÷ mⁿ = 1

...since anything divided by itself is just "1".

Another comment: Please don't ask me to "define" 0⁰. There are at least two ways of looking at this quantity:

• Anything to the zero power is "1", so 0⁰ = 1.
• Zero to any power is zero, so 0⁰ = 0.

As far as I know, the "math gods" have not yet settled on a "definition" of 0⁰. In fact, in calculus, "0⁰" will be called an "indeterminate form". If this quantity comes up on class, don't assume: ask your instructor what you should do with it.

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