Grade by Grade
B. Exponential Functions · Exponential Functions

Zero and Negative Exponents with Fraction, Decimal and Negative Bases

B1.2 B1.5About 70 minutes
Learning goals
  • I can explain why a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0) using patterns and the quotient rule.
  • I can evaluate powers with integer exponents and rational bases, such as (2/3)⁻², 0.5⁻³ and (−4)⁻².
  • I can tell apart (−2)⁻², −2⁻² and 2⁻² without a calculator.
  • I can simplify and evaluate numerical expressions with integer exponents, by hand and with a calculator.

Words to know

zero exponent
Any non-zero base raised to the exponent 0 equals 1: a⁰ = 1, a ≠ 0.
negative exponent
A negative exponent means the reciprocal of the power: a⁻ⁿ = 1/aⁿ, a ≠ 0.
reciprocal
The number you multiply by to get 1. The reciprocal of 2/3 is 3/2; the reciprocal of 5 is 1/5.
rational number
A number that can be written as a fraction a/b of integers, b ≠ 0. Examples: 3, −0.25, 2/7.
integer exponent
An exponent that is a whole number or its opposite: …, −2, −1, 0, 1, 2, …

Going backwards in time

A population of bacteria in a lab dish doubles every hour. Right now there are 1000. In 1 hour there will be 1000 × 2¹ = 2000, and in 3 hours, 1000 × 2³ = 8000. What about 2 hours ago? Going back one hour halves the count, so 2 hours ago there were 1000 ÷ 2 ÷ 2 = 250. Using the same formula with t = −2 gives 1000 × 2⁻² = 1000 × 1/4 = 250. It works!

Negative and zero exponents are what let exponential models run backward in time. You met them in Grade 10; now you will use them with fraction, decimal and negative bases too.

Why a⁰ = 1 and a⁻ⁿ = 1/aⁿ

Pattern: each time the exponent goes down by 1, you divide by the base.

3³ = 27, 3² = 9, 3¹ = 3, 3⁰ = 3 ÷ 3 = 1, 3⁻¹ = 1 ÷ 3 = 1/3, 3⁻² = 1/3 ÷ 3 = 1/9.

Quotient rule: 5² ÷ 5² is 25 ÷ 25 = 1, but the rule says 5²⁻² = 5⁰. So 5⁰ must be 1. And 5² ÷ 5⁴ = 25/625 = 1/25 = 1/5², but the rule says 5⁻². So 5⁻² = 1/5².

The two definitions (for any a ≠ 0):

  • a⁰ = 1
  • a⁻ⁿ = 1/aⁿ, and also 1/a⁻ⁿ = aⁿ

A negative exponent does not make the answer negative. It makes it a reciprocal.

1/82⁻³1/42⁻²1/22⁻¹12⁰22¹42²82³negative exponents:small but still positive
Powers of 2 from 2⁻³ to 2³. Each bar is half the one to its right. Negative exponents give small positive numbers, never negative ones.

Fraction bases: flip it

For a fraction base, the reciprocal is just the fraction turned upside down:

(a/b)⁻ⁿ = (b/a)ⁿ

Why? (2/3)⁻² = 1 ÷ (2/3)² = 1 ÷ 4/9 = 9/4, and (3/2)² is also 9/4.

Decimals work the same way once you write them as fractions: 0.5 = 1/2, so 0.5⁻³ = (1/2)⁻³ = 2³ = 8. Similarly 0.2 = 1/5, so 0.2⁻² = 5² = 25.

(2/3)⁻²flip the baseexponent → +2(3/2)²9/4A negative exponent means "take the reciprocal": 1 ÷ (2/3)² = 1 ÷ 4/9 = 9/4.
To evaluate a fraction to a negative exponent, flip the fraction and make the exponent positive.

Negative bases and the minus-sign trap

Brackets decide what the base is. Compare:

  • (−2)⁻² = 1/(−2)² = 1/4 (the base is −2; an even power is positive)
  • (−2)⁻³ = 1/(−2)³ = 1/(−8) = −1/8 (odd power of a negative is negative)
  • −2⁻² = −(2⁻²) = −1/4 (the base is 2; the minus sign is applied last)
  • 2⁻² = 1/4

Order of operations (BEDMAS) puts exponents before multiplication, so the minus sign in −2⁻² waits until the power is done.

Worked example

Example 1: Evaluate powers with rational bases

Evaluate without a calculator: (a) (3/4)⁻² (b) (−1/2)⁻⁵ (c) 0.1⁻³ (d) (−7)⁰ (e) −5⁰

  1. (a) Flip and make the exponent positive: (4/3)² = 16/9.
  2. (b) Flip: (−2/1)⁵ = (−2)⁵ = −32 (odd power of a negative).
  3. (c) 0.1 = 1/10, so 0.1⁻³ = 10³ = 1000.
  4. (d) Any non-zero base to the 0 is 1: (−7)⁰ = 1.
  5. (e) The base is 5: −5⁰ = −(5⁰) = −(1) = −1.

16/9, −32, 1000, 1, −1

Worked example

Example 2: Simplify first, then evaluate

Evaluate (2⁻³ × 2⁷) ÷ 2⁶.

  1. The rules work with negative exponents too. Top: 2⁻³⁺⁷ = 2⁴.
  2. Divide: 2⁴⁻⁶ = 2⁻².
  3. 2⁻² = 1/2² = 1/4.
  4. Calculator check: enter (2^(−3) × 2^7) ÷ 2^6. The display shows 0.25 = 1/4. ✓

1/4 (0.25)

Worked example

Example 3: Add powers (no rule shortcut!)

Evaluate 4⁻¹ + 2⁻³ − 3⁰.

  1. This is a sum, so there is no exponent rule to combine the powers. Evaluate each power first (order of operations).
  2. 4⁻¹ = 1/4, 2⁻³ = 1/8, 3⁰ = 1.
  3. 1/4 + 1/8 − 1 = 2/8 + 1/8 − 8/8 = −5/8.

−5/8 (−0.625)

Calculator tips

  • Use the power key (it may be labelled xʸ, yˣ or ^) and the negative key (−), not the subtraction key, for negative exponents: 2 ^ (−) 3 = 0.125.
  • Put fraction bases in brackets: (2 ÷ 3) ^ (−) 2 = 2.25 = 9/4. Without brackets, 2 ÷ 3^(−2) gives 18, which is a different expression.
  • For a negative base type the brackets yourself: (−2)^4 = 16, but −2^4 = −16 on most calculators.
  • Many calculators have a fraction key (often a b/c) that turns 0.5625 into 9/16. Exact fraction answers are best when the question gives fractions.
Just for fun

Why can't you ever divide by zero, even with negative exponents?

Because 0⁻¹ would be 1/0, and that is one fraction even the strongest calculator refuses to lift. (Try it: your calculator will show an error message.)

Hands-on activity

Halving paper strips backward

You need: a long strip of paper (about 30 cm), a ruler, scissors (take care; a grown-up can help), and a pen.

  1. Call the full strip length 1 = 2⁰ strip. Write "2⁰ = 1" on it.
  2. Fold the strip in half and cut. One piece is 1/2 of the strip. Label it "2⁻¹ = 1/2".
  3. Halve a piece again: label "2⁻² = 1/4". Again: "2⁻³ = 1/8". Keep going as long as you can cut safely.
  4. Measure each piece with the ruler and record the lengths in a table with the exponent.
  5. Now tape two full strips end to end: that is 2¹ = 2 strips long. Four strips would be 2².

What to look for: with a 30 cm strip the pieces are about 15 cm, 7.5 cm, 3.75 cm and 1.9 cm. Each step down in the exponent halves the length, and no piece is ever a negative length. A negative exponent means a smaller piece, not a negative one.

Your turn

1

Evaluate: 6⁻²

2

Evaluate: (2/5)⁻³

3

Evaluate: 0.25⁻²

4

Evaluate: (−3)⁻³

5

Evaluate: −4⁻²

6

Evaluate: (5/8)⁰ + 8⁰

7

Simplify, then evaluate: 3⁻⁴ × 3⁶

8

Simplify, then evaluate: 10³ ÷ 10⁵

More practice: warm-up

1

Evaluate: 9⁰

2

Evaluate: 2⁻¹

3

Evaluate: 10⁻³

4

Evaluate: (1/3)⁻²

5

True or false: 5⁻² is a negative number.

More practice: on your own

1

Evaluate: (−2/3)⁻⁴

2

Evaluate: 2⁻¹ + 3⁻¹

3

Simplify, then evaluate: (4⁻² × 4⁵) ÷ (4²)²

4

Evaluate: 1.5⁻² (as a fraction)

5

Put in order from least to greatest: 2⁻³, (−2)⁻³, −2³, (1/2)⁻³

Challenge

1

Evaluate: [(2/3)⁻² − (3/2)⁰]⁻¹

2

Find the integer n if 3ⁿ = 1/81.

3

A bacteria count doubles every hour and is 6400 now. Use a power of 2 with a negative exponent to find the count 5 hours ago.

Check your understanding

Quiz

1.

What is the value of (4/7)⁰?

2.

Evaluate (1/5)⁻².

3.

Which expression has a negative value?

4.

Simplify 7⁻³ × 7⁵ ÷ 7².

5.

Evaluate 0.5⁻⁴.

6.

Which is equal to −3⁻²?

For parents and teachers

Big idea (B1.2, B1.5): students evaluate expressions with integer exponents and rational (fraction, decimal, negative) bases, and use the exponent rules with negative and zero exponents.

How to help: Ask, "What does the negative exponent tell you to do?" The answer should be "take the reciprocal", not "make it negative". When your teen gets a calculator answer, ask them to predict the sign and rough size first.

Common difficulties:

  • Thinking a⁻ⁿ is negative: 5⁻² = 1/25, not −25.
  • Forgetting to flip the fraction: (2/3)⁻² = 9/4, not 4/9.
  • Brackets on negative bases: (−2)⁻² = 1/4 but −2⁻² = −1/4. On the calculator, the brackets must be typed.
  • Combining sums with exponent rules: 2⁻¹ + 3⁻¹ is 5/6, not 5⁻¹.
  • Saying 0⁰ = 1: the definition a⁰ = 1 needs a ≠ 0. Students do not need to evaluate 0⁰ or 0⁻ⁿ.