Extending exponents beyond the counting numbers. Includes deriving that a nonzero base to the zero power must be one and that a negative exponent means a reciprocal, both forced by the product of powers rule, evaluating such powers, distinguishing a negative exponent from a negative value, and applying the Lesson 8.1 rules unchanged.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 8 — Exponents and Exponential Functions
Zero and Negative Exponents
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-454 — the lesson these objectives are drawn from
Warm-up
Lesson 8.1 defined exponents by counting factors, which only makes sense for positive whole numbers. This lesson asks what the other exponents must mean.
Discussion prompt
The product rule says a to the zero times a to the n is a to the n. What does that force a to the zero to be?
Hint: Something multiplied by a to the n gives a to the n back.
Answer:
\[ a^0 \cdot a^n = a^{0 + n} = a^n \]
Only one number leaves a quantity unchanged when multiplied by it, and that is one. So a to the zero must be one — the rule leaves no choice, and the definition is forced rather than invented.
Concept
Counting factors gives no meaning to an exponent of zero or a negative exponent. Requiring the product of powers rule to keep working forces exactly one value for each, and those forced values become the definitions.
negative exponent — A negative exponent means the reciprocal of the corresponding positive power: a to the negative n is one over a to the n, for a nonzero.
Zero to the zero power is left undefined.
Figure (svg): The product rule used to force the value of a zero exponent
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-449
Section
Section 1
Concept
Applying the product of powers rule to a to the zero times a to the n gives a to the n. Something that leaves a quantity unchanged must be one, so a to the zero is one for any nonzero base.
\[ a^0 \cdot a^n = a^n \;\Longrightarrow\; a^0 = 1, \quad a \neq 0 \]
The base has to be nonzero, since the argument divides by a to the n.
Figure (svg): The product rule used to force the value of a zero exponent
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-449 — the derivation of a to the zero from the product of powers property
Picture it
Forced, not chosen.
Figure (svg): The product rule used to force the value of a zero exponent
The definition is the only one that keeps Lesson 8.1's rules true. Any other value would break the product rule at the first opportunity.
Worked example
This is Example 1 from the textbook.
\[ \text{Evaluate } \; 5^0, \quad 0^0, \quad (-2)^0, \quad \left(\tfrac{1}{9}\right)^0. \]
Take the first
Why: Five is nonzero, so the result is one.
\[ 1 \]
Take the second
Why: The definition excludes a base of zero.
Take the third
Why: Negative two is nonzero.
\[ 1 \]
Take the fourth
Why: A ninth is nonzero.
\[ 1 \]
Figure (svg): Several bases raised to the zero power, all giving one
\[ 1, \quad \text{undefined}, \quad 1, \quad 1 \]
Verify: notice what the three answers have in common
Why: The base made no difference at all — positive, negative or fractional, the answer was one. That is the point of the definition, and it is why only the excluded base of zero behaves differently.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-449
Sorting
Only one base is excluded.
Sort into buckets
Sort each expression by its value.
Five of the six are one, and the sixth is the single exception. The base's size, sign and type make no difference at all as long as it is not zero.
Worked example
A second route to the same value.
\[ \text{List } \; 2^3, 2^2, 2^1 \; \text{ and continue the pattern to } 2^0. \]
List the powers
Why: Eight, four, two.
\[ 8, 4, 2 \]
Describe the pattern
Why: Each is half the one before it.
Continue one step
Why: Half of two is one.
\[ 2 ^{0} = 1 \]
Compare with the algebra
Why: The product rule forced the same value.
Figure (svg): A table of powers of two continued downwards through zero into negatives
\[ 8, \; 4, \; 2, \; 1 \]
Verify: check the pattern makes sense in reverse
Why: Going up the list doubles each time, which is what multiplying by another factor of two does. Going down divides by two, and one step below two to the first is two to the zero — so the pattern is just the definition of a power, extended by one line.
Trap
\[ 5^0 \]
Answer zero, since the exponent is zero
Why: The exponent's value seems like it should be the answer's value.
Multiplying zero factors of five leaves you with nothing removed rather than nothing left. The product rule forces the answer to be one, since one is what leaves other powers unchanged.
\[ 5^0 = 1 \]
Ask what value keeps the product rule true
Why: Five to the zero times five squared has to be five squared, and only one does that.
The halving pattern gives the same answer independently, which is worth seeing once.
Faded example
The product rule decides it.
Fill in the blanks
Since a^0 times a^n equals a^n, the value a^0 must be 1, provided the base is not 0.
Why: Only one leaves a product unchanged, so the rule forces that value. The exclusion of zero comes from the argument needing to divide by a to the n, which is zero when the base is zero.
Elimination
Every other base gives one.
Eliminate the wrong options
What is the reason?
Survives elimination: A
Why: The derivation divides both sides by a to the n, which is legitimate only when the base is nonzero. With a base of zero there is nothing to divide by, so no value is forced and none is assigned.
Socratic
The definitions could have been chosen differently.
Discussion prompt
Explain why mathematicians define a to the zero as one rather than choosing some other value. Then say what would break if it were defined as zero instead.
Hint: Ask what the definition is meant to preserve.
Answer:
The definition is chosen so that the rules already established keep working. Exponents were defined by counting factors, which says nothing about an exponent of zero, so a value has to be assigned — and the only sensible criterion is that the product rule should continue to hold.
Defining it as zero would break that immediately: five to the zero times five squared would be zero times twenty-five, which is zero, and the product rule says it should be five squared. Every later rule that relies on the product rule would then fail too, so the choice is not free — it is forced by what has already been built.
Section
Section 2
Concept
Applying the product rule to a to the negative n times a to the n gives a to the zero, which is one. Something multiplying a to the n to give one is its reciprocal.
\[ a^{-n} \cdot a^n = a^0 = 1 \;\Longrightarrow\; a^{-n} = \dfrac{1}{a^n} \]
The same exclusion of zero applies, since zero has no reciprocal.
Figure (svg): The product rule used to force the meaning of a negative exponent
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-449 — the derivation of a negative exponent and the Zero and Negative Exponents box
Picture it
The product must be one.
Figure (svg): The product rule used to force the meaning of a negative exponent
The definition can also be written the other way round: one over a to the negative n is a to the n. Both forms say that a negative exponent moves a factor between numerator and denominator.
Worked example
This is Example 2 from the textbook.
\[ \text{Evaluate } \; 2^{-2} \; \text{ and } \; (-3)^{-4}. \]
Take the first
Why: The reciprocal of two squared.
\[ 1 / 2 ^{2} \]
Evaluate
Why: One quarter.
\[ \frac{1}{4} \]
Take the second
Why: The reciprocal of negative three to the fourth.
\[ 1 / (-3) ^{4} \]
Evaluate
Why: Negative three to the fourth is eighty-one.
\[ \frac{1}{81} \]
Figure (svg): A negative exponent evaluated in two steps
\[ \tfrac{1}{4} \qquad \tfrac{1}{81} \]
Verify: check that both answers are positive
Why: A negative exponent produces a reciprocal, and a reciprocal of a positive number is positive. The second base was negative and its fourth power is positive, so the answer is positive too — the exponent's sign never determines the answer's sign.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 450-450
Faded example
Reciprocate first, evaluate second.
Fill in the blanks
2^2 = \dfrac4___}}} = \dfrac______}
Why: The negative exponent becomes a positive one in the denominator, and evaluating that gives four. Doing the reciprocal and the power as separate steps is what keeps the two ideas from being mixed.
Worked example
Guided Practice 1 to 4. One zero exponent and three negative ones.
\[ \text{Evaluate } \; 18^0, \quad (-9)^{-2}, \quad 2^{-3}, \quad (-5)^{-2}. \]
Take the first
Why: A nonzero base to the zero power.
\[ 1 \]
Take the second
Why: The reciprocal of negative nine squared, which is eighty-one.
\[ \frac{1}{81} \]
Take the third
Why: The reciprocal of eight.
\[ \frac{1}{8} \]
Take the fourth
Why: The reciprocal of twenty-five.
\[ \frac{1}{25} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 1, \quad \tfrac{1}{81}, \quad \tfrac{1}{8}, \quad \tfrac{1}{25} \]
Verify: check one by multiplying
Why: Two to the negative three is an eighth, and an eighth times two cubed is an eighth times eight, which is one. Every power and its negative multiply to one, so that product is the quickest confirmation available.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 450-450
Error analysis
The student evaluated two powers with negative exponents.
Annotate
On: \( \begin{aligned} 2^{-2} &= -4 \\ (-3)^{-4} &= -\tfrac{1}{81} \end{aligned} \)
Multiplying each answer by the corresponding positive power settles it: one quarter times four is one, and negative four times four is negative sixteen.
Translation
Reciprocal of the positive power.
Match the pairs
Why: Every answer is a positive fraction, because a reciprocal of a positive number is positive and each base's even power is positive. The negative sign in the exponent never reaches the value's sign.
Elimination
The exponent is negative and the base is positive.
Eliminate the wrong options
What is the effect on the value?
Survives elimination: A
Why: The definition makes it the reciprocal of the positive power, which for a base greater than one is a proper fraction. Option B is the standard confusion and it is caught by multiplying: a quarter times four is one, while negative four times four is not.
Socratic
The value is forced, like the last one.
Discussion prompt
Explain why the product rule forces a negative exponent to mean a reciprocal. Then say why a base of zero has to be excluded here too.
Hint: Ask what the two powers must multiply to.
Answer:
Adding negative n and n gives zero, so the product of the two powers is a to the zero, which the previous section forced to be one. A number whose product with a to the n is one is by definition the reciprocal of a to the n, so the value is determined.
Zero has no reciprocal, since nothing multiplied by zero gives one. So a base of zero is excluded from the negative-exponent definition for the same kind of reason it was excluded from the zero-exponent one — the forcing argument requires something that does not exist there.
Section
Section 3
Concept
A negative exponent makes a number small and leaves its sign alone. A minus sign in front of a power makes the result negative and leaves its size alone.
\[ 2^{-2} = \tfrac{1}{4} \qquad -2^2 = -4 \]
The sign of a power comes from the base and the parity of the exponent, never from the exponent's sign.
Figure (svg): Two columns contrasting a negative exponent with a negative value
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 450-450 — Example 2, where both answers are positive
Picture it
One reciprocates; one negates.
Figure (svg): Two columns contrasting a negative exponent with a negative value
The left column produces a quarter and the right produces negative four. The two expressions differ by where the minus sign sits, and by a factor of sixteen.
Worked example
Position of the minus sign decides everything.
\[ \text{Evaluate } \; 2^{-2}, \quad -2^2, \quad (-2)^2, \quad (-2)^{-2}. \]
Take the first
Why: A negative exponent, so a reciprocal.
\[ \frac{1}{4} \]
Take the second
Why: The minus applies after squaring.
\[ -4 \]
Take the third
Why: The base is negative and the exponent even.
\[ 4 \]
Take the fourth
Why: Reciprocal of the third.
\[ \frac{1}{4} \]
Figure (svg): Two columns contrasting a negative exponent with a negative value
\[ \tfrac{1}{4}, \quad -4, \quad 4, \quad \tfrac{1}{4} \]
Verify: compare the first and the fourth
Why: Both give a quarter, because negative two squared is four just as two squared is, and the reciprocal of four is a quarter either way. The base's sign disappeared under an even exponent, which is what makes the two agree.
Sorting
The base and the parity decide the sign.
Sort into buckets
Sort each expression by the sign of its value.
Not one of these signs was decided by whether the exponent was negative. That sign controls only whether the answer is a whole number or a fraction.
Worked example
The base and the parity, never the exponent's sign.
\[ \text{Decide the sign of } \; (-3)^{-4} \; \text{ and } \; (-3)^{-3}. \]
Take the first
Why: The fourth power of a negative is positive.
Reciprocate
Why: The reciprocal of a positive is positive.
\[ \frac{1}{81} \]
Take the second
Why: The third power of a negative is negative.
Reciprocate
Why: The reciprocal of a negative is negative.
\[ -\frac{1}{27} \]
Figure (svg): The solution to Worked example where does the sign come from shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{1}{81} \qquad -\tfrac{1}{27} \]
Verify: say which feature decided each sign
Why: The parity of the exponent decided both: even gave a positive result and odd a negative one. The minus sign in the exponent affected only whether the answer was a whole number or a fraction, which is the distinction the whole section is about.
Trap
\[ 5^{-2} \]
Write negative twenty-five, since the exponent is negative
Why: A minus sign somewhere in the expression seems to belong in the answer.
The exponent's sign is an instruction to reciprocate, so the answer is one twenty-fifth. Negative twenty-five is not even close: it is six hundred and twenty-five times too large in size, and the wrong sign.
\[ 5^{-2} = \tfrac{1}{5^2} = \tfrac{1}{25} \]
Ask where each minus sign is and what it does
Why: In the exponent it reciprocates; in front of the base or the whole power it negates.
Multiplying the answer by five squared is the check: a twenty-fifth times twenty-five is one, as it must be.
Faded example
Position changes the meaning.
Fill in the blanks
2^(-2) equals 1/4 because the exponent reciprocates, while -2^2 equals -4 because the minus sign negates the result.
Why: The two expressions share the same digits and differ only in where the minus sign sits, and their values differ by a factor of sixteen and a sign. Reading the position of the minus before evaluating is what tells them apart.
Prediction
A power with a negative base and a negative exponent.
Predict first
What determines whether (-4)^(-3) is positive or negative?
Correct: The parity of the exponent: odd gives a negative result.
\[ (-4)^{-3} = \dfrac{1}{(-4)^3} = -\tfrac{1}{64} \]
Why: Negative four cubed is negative sixty-four, and the reciprocal of a negative number is negative, so the answer is negative one sixty-fourth. The exponent's sign controls only the reciprocation, and reciprocating preserves whatever sign was already there. An even exponent would have made both the power and its reciprocal positive.
Socratic
Two minus signs doing different jobs.
Discussion prompt
Explain why a negative exponent cannot change whether the answer is positive or negative. Then say what it does change, and give an example where the two effects are easy to confuse.
Hint: Ask what reciprocating does to a sign.
Answer:
A negative exponent means take the reciprocal, and one divided by a positive number is positive while one divided by a negative is negative. So reciprocating preserves the sign exactly, and the sign was already fixed by the base and the parity before the reciprocation happened.
What it changes is the size: a base larger than one gives a value smaller than one, and vice versa. The pair 2 to the negative two and negative two squared is the confusing case, since both contain a two and a minus sign and give a quarter and negative four — the same digits, and answers differing by a factor of sixteen with opposite signs.
Section
Section 4
Concept
The product of powers, power of a power and power of a product rules all continue to hold when the exponents are zero or negative. The arithmetic on the exponents is now done with integers.
The definitions were chosen precisely so that this would be true.
Figure (svg): The multiplication rules applied to expressions with negative exponents
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 450-450 — Example 3, Evaluate Exponential Expressions
Picture it
Add, then evaluate; multiply, then evaluate.
Figure (svg): The multiplication rules applied to expressions with negative exponents
Both examples produce an exponent outside the counting numbers and then use this lesson's definitions to finish. That two-stage shape is typical of every problem here.
Worked example
This is Example 3 from the textbook.
\[ \text{Evaluate } \; 6^4 \cdot 6^{-4}, \quad (2^3)^{-2}, \quad (3 \cdot 2)^{-2}. \]
Take the first
Why: Add four and negative four to get zero.
\[ 6 ^{0} = 1 \]
Take the second
Why: Multiply three by negative two.
\[ 2 ^{-6} \]
Evaluate it
Why: The reciprocal of sixty-four.
\[ \frac{1}{64} \]
Take the third
Why: Six to the negative two, or a ninth times a quarter.
\[ \frac{1}{36} \]
Figure (svg): The multiplication rules applied to expressions with negative exponents
\[ 1, \quad \tfrac{1}{64}, \quad \tfrac{1}{36} \]
Verify: check the third both ways
Why: Distributing first gives three to the negative two times two to the negative two, which is a ninth times a quarter, or one thirty-sixth. Multiplying first gives six to the negative two, which is one thirty-sixth as well — the power of a product rule holds with negative exponents just as it did with positive ones.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 450-450
Faded example
Add the exponents first.
Fill in the blanks
4^2 \cdot 4^-3 = 4^4})} = 4^___ = \dfrac______}
Why: Two plus negative three is negative one, and four to the negative one is a quarter. Combining first and evaluating once is shorter than computing each power and dividing.
Worked example
Guided Practice 5 to 7. One of each rule.
\[ \text{Evaluate } \; 4^2 \cdot 4^{-3}, \quad (3^{-1})^2, \quad (2 \cdot 5)^{-2}. \]
Take the first
Why: Two plus negative three is negative one.
\[ 4 ^{-1} = \frac{1}{4} \]
Take the second
Why: Negative one times two is negative two.
\[ 3 ^{-2} = \frac{1}{9} \]
Take the third
Why: Ten to the negative two.
\[ \frac{1}{100} \]
Note the pattern
Why: Each rule was applied first and the definition second.
Figure (svg): The solution to Worked example three from guided practice shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{1}{4}, \quad \tfrac{1}{9}, \quad \tfrac{1}{100} \]
Verify: check the first by evaluating both powers
Why: Four squared is sixteen and four to the negative three is one sixty-fourth, and sixteen over sixty-four is a quarter. The rule and the direct computation agree, which is the reassurance the definitions were designed to provide.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 450-450
Trap
\[ 6^4 \cdot 6^{-4} \]
Evaluate both powers separately first: 1296 times 1/1296
Why: Each power can be computed, so computing them looks like progress.
It works and it is far more effort. Adding the exponents gives six to the zero, which is one, in a single line — and for larger exponents the direct route becomes impractical.
\[ 6^4 \cdot 6^{-4} = 6^0 = 1 \]
Combine the exponents first, then evaluate once at the end
Why: That is what the rules are for.
The two-stage shape — rule first, definition second — is what makes these problems short.
Sorting
The structures are the same as in Lesson 8.1.
Sort into buckets
Sort each expression by the property that simplifies it.
The three structures are exactly those of Lesson 8.1. Only the kind of number appearing in the exponent has changed, which is what makes the extension worth having.
Elimination
Add the exponents.
Eliminate the wrong options
Which is the value?
Survives elimination: A
Why: Four plus negative four is zero, and six to the zero is one. Option B is the natural slip: the exponent is zero and the value is not, which is exactly what the first section of this lesson established.
Socratic
They were proved by counting factors.
Discussion prompt
Explain why the multiplication rules continue to hold for zero and negative exponents, even though the counting-factors argument no longer makes sense there. Then say what that says about how the definitions were chosen.
Hint: Ask which came first, the rules or the definitions.
Answer:
The counting argument establishes the rules for positive whole exponents. The definitions for zero and negative exponents were then chosen precisely so that those same rules would continue to hold, so their survival is not a happy accident but the criterion by which the definitions were selected.
It means the whole extension is an act of preservation rather than of invention. Once you decide that the product rule must keep working, the values of a to the zero and a to the negative n are forced, and everything else follows. That pattern — extend a definition by preserving a rule — recurs throughout mathematics, and this is the first place it appears.
Section
Section 5
Concept
A table of powers can be continued downwards by dividing by the base each time. It passes through one at the zero exponent and into fractions for negative exponents, agreeing with the definitions.
The pattern and the algebra give the same answers.
Figure (svg): A table of powers of two continued downwards through zero into negatives
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-454 — the definitions and their use in the lesson's exercises
Picture it
Eight, four, two, one, a half, a quarter.
Figure (svg): A table of powers of two continued downwards through zero into negatives
Nothing in the pattern announces that something new is happening at the exponent zero. The definitions are exactly what makes the sequence continue smoothly.
Worked example
Both routes agree.
\[ \text{Continue } \; 5^3, 5^2, 5^1 \; \text{ down to } 5^{-2}. \]
List the known powers
Why: One hundred and twenty-five, twenty-five, five.
\[ 125, 25, 5 \]
Divide by five to continue
Why: Five divided by five is one.
\[ 5 ^{0} = 1 \]
Continue
Why: One divided by five is a fifth.
\[ 5 ^{-1} = \frac{1}{5} \]
And again
Why: A fifth divided by five is a twenty-fifth.
\[ 5 ^{-2} = \frac{1}{25} \]
Figure (svg): A table of powers of two continued downwards through zero into negatives
\[ 5^0 = 1, \; 5^{-1} = \tfrac{1}{5}, \; 5^{-2} = \tfrac{1}{25} \]
Verify: compare with the definitions
Why: The definitions give five to the zero as one and five to the negative two as the reciprocal of twenty-five, matching the pattern exactly. Two independent routes agreeing is what makes the definitions feel inevitable rather than imposed.
Faded example
Divide by the base each step.
Fill in the blanks
The powers 8, 4, 2 continue to 2^0 = 1, then 2^(-1) = 1/2.
Why: Each step down halves the previous value, and the sequence passes through one at the zero exponent without interruption. That the pattern and the algebra give the same answers is what makes the definitions natural.
Worked example
Exercise 64 uses such a model for population.
\[ \text{A quantity is modelled by } \; P = 250 \cdot 1.02^t. \text{ What does } t = -20 \text{ give?} \]
Interpret the negative input
Why: Twenty units before the reference time.
\[ 20\text{ earlier} \]
Write the power
Why: 1.02 to the negative twenty.
\[ 1.02 ^{-20} \]
Use the definition
Why: The reciprocal of 1.02 to the twentieth.
\[ 1 / 1.02 ^{20} \]
Interpret the result
Why: Dividing by the growth undoes it, giving a smaller earlier value.
Figure (svg): The solution to Worked example a model with a negative exponent shown as a ladder of expressions, one row per algebraic move
\[ 1.02^{-20} = \dfrac{1}{1.02^{20}} \]
Verify: say why the answer should be smaller
Why: The quantity grows by two per cent each unit of time, so going back in time should give a smaller figure. The reciprocal does exactly that, which is why negative exponents let a growth model run backwards — Lesson 8.6 develops this.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 454-454
Trap
\[ P = 250 \cdot 1.02^{-20} \]
Expect a negative population, since the exponent is negative
Why: A negative input suggests a negative output.
The power is a reciprocal, so it is a positive number less than one, and the population comes out smaller than two hundred and fifty rather than negative. A model of a count never produces a negative value from a negative exponent alone.
A negative exponent gives a value between zero and one, so the model produces a smaller positive quantity.
Read a negative exponent as running the process backwards
Why: Dividing by the growth factor repeatedly, rather than multiplying by it.
Checking the direction against the situation is the same interpretation step as in Lesson 5.5, applied to an exponential model.
Prediction
The base is greater than one.
Predict first
Where does 3^(-4) sit compared with 1?
Correct: Between 0 and 1.
\[ 3^{-4} = \tfrac{1}{81} \approx 0.012 \]
Why: Three to the fourth is eighty-one, so the reciprocal is one eighty-first — positive and less than one. Any base greater than one raised to a negative exponent lands strictly between zero and one, which is a useful sanity check on an answer.
Hypothesis
Predict before you check.
Predict first
What is (1/2)^(-3)?
Correct: 8, since the reciprocal of a fraction is larger than 1.
\[ \left(\tfrac{1}{2}\right)^{-3} = \dfrac{1}{(1/2)^3} = \dfrac{1}{1/8} = 8 \]
Why: A half cubed is an eighth, and the reciprocal of an eighth is eight. So a base below one with a negative exponent gives a value above one, which is the reverse of what happens with a base above one — the negative exponent reciprocates, and reciprocating a small number gives a large one.
Socratic
The definitions could have been left out.
Discussion prompt
Say what negative exponents make possible that positive ones alone do not. Then name two places later in this chapter where they will be needed.
Hint: Think about very small numbers and about running a process backwards.
Answer:
They let a single notation cover both large and small numbers, and they let an exponential model run backwards in time. Without them, dividing by a power would need a different notation from multiplying by one, and the rules would have to be stated twice.
They are needed for scientific notation in Lesson 8.5, where a number like 0.00034 is written as 3.4 times ten to the negative four, and for exponential decay in Lesson 8.7, where repeated shrinking is described by the same machinery as repeated growth. Both would be far clumsier without the extension made here.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Exponent | Meaning | Value for a base above 1 |
|---|---|---|
| positive n | n factors of the base | greater than 1 |
| zero | forced by the product rule | exactly 1 |
| negative n | the reciprocal of a to the n | between 0 and 1 |
Reading the table downwards, the values shrink steadily through one and into fractions. That is the pattern the definitions were chosen to preserve.
Pattern
Whether the exponents are positive, zero or negative, the same five moves cover it.
Step three before step five is deliberate: reciprocating first and evaluating second keeps the two ideas separate, which is where the sign confusions come from.
OpenStax Elementary Algebra 2e, §6.7 Integer Exponents and Scientific Notation §6.7
Check
The base is what matters.
Check your understanding
What is (-7)^0?
Answer: A
Why: Negative seven is nonzero, so the definition gives one. The base's sign makes no difference; only a base of zero is excluded.
Check
Reciprocate, then evaluate.
Check your understanding
What is 3^(-2)?
Answer: A
Why: The negative exponent gives the reciprocal of three squared, which is one ninth. Multiplying by nine gives one, confirming it.
Check
Combine, then evaluate.
Check your understanding
What is 5^3 times 5^(-3)?
Answer: A
Why: Three plus negative three is zero, and five to the zero is one. Directly, one hundred and twenty-five times one over one hundred and twenty-five is also one.
Real world
This is Exercise 64's situation. A model estimates the United States population as 250 million times 1.02 raised to t, where t is the number of years after 1990.
Discussion prompt
Work out what t would represent the year 1776, write the corresponding power using the definition, and say what kind of number it is. Then say what the model's answer should be treated as.
Hint: 1776 is before the reference year.
Answer:
\[ t = 1776 - 1990 = -214 \;\Longrightarrow\; P = 250 \cdot 1.02^{-214} = \dfrac{250}{1.02^{214}} \]
The negative exponent gives a reciprocal, so the power is a small positive number and the population comes out far below two hundred and fifty million — which is the right direction, since the country was much smaller then.
The figure should be treated as an extrapolation rather than an estimate. The model was fitted to modern growth, and running it back two hundred and fourteen years assumes a constant two per cent rate across a period of wars, immigration waves and epidemics. Negative exponents let the model run backwards; whether the answer means anything is a separate question, and here it does not mean much.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
What is 2^(-3)?
Correct: 1/8, since a negative exponent means a reciprocal.
\[ 2^{-3} = \dfrac{1}{2^3} = \tfrac{1}{8} \]
\[ \tfrac{1}{8} \cdot 2^3 = 1 \;\checkmark \]
Why: The definition makes it one over two cubed, which is an eighth — small and positive. The first option is the standard error, treating the exponent's minus sign as a sign on the answer, and it is caught immediately by multiplying: an eighth times eight is one, as the definition requires, while negative eight times eight is negative sixty-four. A negative exponent makes a number small; a minus sign in front makes it negative.
Explain it
They think a negative exponent makes the answer negative.
Discussion prompt
In no more than four sentences, explain what a negative exponent means and why the answer is not negative. Then give them a check they can do in their head.
Hint: Reciprocal, not minus.
Answer:
A usable answer: a negative exponent tells you to flip the number over rather than to make it negative — two to the negative three means one over two cubed, which is an eighth. It makes a number small, not negative, and the sign of the answer comes from the base, exactly as it always did.
The check is to multiply your answer by the same power with a positive exponent: an eighth times eight is one, which is what the definition says has to happen. If you get anything other than one, the exponent was mishandled.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The zero exponent is fixed by asking what value keeps the product rule true, which is one. The reciprocal reading is fixed by multiplying your answer by the positive power and checking you get one. Signs are fixed by remembering that the base and the parity decide them, never the exponent's sign. Applying the rules is fixed by doing the exponent arithmetic first and the definitions second. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page derive both definitions in two lines each: write a to the zero times a to the n and use the product rule to force the value, then do the same for a to the negative n. Underneath, build a table of powers of three from the exponent four down to the exponent negative two, writing the value at each step and the division that produced it, and circling the entry at the exponent zero. In the middle, evaluate four expressions with negative or zero exponents, showing the rule you used and then the definition, and beside each write the product of your answer with the corresponding positive power to check that it is one. In the lower half, write the four expressions two to the negative two, minus two squared, negative two squared and negative two to the negative two, evaluate all four, and write one sentence on what distinguishes them. Finally, in the margin, write which base is excluded and why.
Every one of your check products should be exactly one. If one is not, that answer's exponent was mishandled — most likely the reciprocal was taken of the wrong thing.
Recap
Five things, and the third is the one that is easiest to get backwards.
| If the question says | Your first move is |
|---|---|
| A nonzero base to the zero power | Write 1 |
| A negative exponent | Write the reciprocal of the positive power |
| Zero to the zero power | Say it is undefined |
| Two powers with the same base | Add the exponents, negatives included |
| Check a negative-exponent answer | Multiply it by the positive power |
Lesson 8.3 leaves arithmetic and draws these powers. Plotting an exponential function shows how it differs from every line in Chapters 4 and 5, and negative exponents are what give it a left-hand half.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.2 Zero and Negative Exponents §8.2, pp. 449-454 — everything on these slides traces back here
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