Grade by Grade
B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Powers: How the Sign and Size of the Exponent Change the Value

B2.1 A1About 70 minutes
Learning goals
  • I can identify the base and exponent of a power and evaluate powers with positive, zero and negative integer exponents.
  • I can use a pattern to explain why a⁰ = 1 and why a⁻ⁿ = 1/aⁿ.
  • I can predict the sign of a power with a negative base, and tell (−3)² apart from −3².
  • I can explain how the size of a power changes as the exponent grows, for bases greater than 1 and bases between 0 and 1.

Words to know

power
An expression like 2⁵ that shows repeated multiplication of the same factor.
base
The factor being multiplied. In 2⁵ the base is 2.
exponent
How many times the base is used as a factor. In 2⁵ the exponent is 5.
zero exponent
Any non-zero base with exponent 0 equals 1: a⁰ = 1 (a ≠ 0).
negative exponent
A negative exponent means the reciprocal: a⁻ⁿ = 1/aⁿ (a ≠ 0).
reciprocal
1 divided by a number. The reciprocal of 5 is 1/5; the reciprocal of 2/3 is 3/2.

Remember from Grade 8

In Grade 8 you used exponents in the order of operations, and you met negative exponents with base 10: 10⁻¹ = 0.1 and 10⁻³ = 0.001. In Grade 9 you will use any base, including negative numbers and fractions, and explain why the rules work.

Doubling bacteria

A lab sample has 500 bacteria at noon, and the number doubles every hour. At 1 p.m. there are 500 × 2 = 1000, at 2 p.m. 500 × 2² = 2000, at 3 p.m. 500 × 2³ = 4000. The formula is 500 × 2ⁿ, where n is the number of hours after noon.

What about 11 a.m. (n = −1) or 10 a.m. (n = −2)? Going back in time means halving: 250 and then 125. So 500 × 2⁻¹ should be 250, and 500 × 2⁻² should be 125. That means 2⁻¹ = 1/2 and 2⁻² = 1/4. Negative exponents are not strange; they are the pattern continued.

Parts of a power

In 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, 2 is the base, 5 is the exponent, and 2⁵ is the power. Read it as "2 to the exponent 5" or "2 to the fifth".

The pattern behind zero and negative exponents

Each time the exponent goes down by 1, you divide by the base once more.

2³= 8÷ 22²= 4÷ 22¹= 2÷ 22⁰= 1÷ 22⁻¹= 1/2÷ 22⁻²= 1/4
Going down the exponents, each value is the previous one divided by 2. Following the pattern past 2¹ gives 2⁰ = 1, 2⁻¹ = 1/2 and 2⁻² = 1/4. Bars are drawn to scale.
Power3⁴3³3²3¹3⁰3⁻¹3⁻²3⁻³
Value81279311/31/91/27
The same pattern with base 3: divide by 3 each step to the right.

The two rules

  • Zero exponent: a⁰ = 1, for any a ≠ 0. (5⁰ = 1, (−7)⁰ = 1, (2/3)⁰ = 1.)
  • Negative exponent: a⁻ⁿ = 1/aⁿ, for any a ≠ 0. (4⁻² = 1/4² = 1/16.)

A negative exponent does not make the value negative. 2⁻³ = 1/8, which is positive. It means "take the reciprocal".

0⁰ is not defined in this course, and 0 to a negative exponent is undefined because it would mean dividing by 0.

The sign of the base

Watch what happens with a negative base:

  • (−2)¹ = −2
  • (−2)² = (−2)(−2) = 4
  • (−2)³ = (−2)(−2)(−2) = −8
  • (−2)⁴ = 16

Each extra factor of −2 flips the sign. An even exponent gives a positive value, and an odd exponent gives a negative value.

Brackets matter. In (−3)², the base is −3, so (−3)² = 9. In −3², the base is just 3, and the negative sign is applied after: −3² = −(3 × 3) = −9.

-8-40481216
Powers of −2 bounce from side to side of zero: (−2)¹ = −2, (−2)² = 4, (−2)³ = −8, (−2)⁴ = 16. Each step multiplies by −2: the size doubles and the sign flips.

The size of the power

  • If the base is greater than 1, a bigger exponent gives a bigger value: 2¹ < 2² < 2³ < … , and negative exponents give values between 0 and 1.
  • If the base is between 0 and 1, a bigger exponent gives a smaller value: (1/2)¹ = 1/2, (1/2)² = 1/4, (1/2)³ = 1/8. Each time you take half of a half.
  • A negative exponent on a fraction flips the fraction: (2/3)⁻² = (3/2)² = 9/4.
Worked example

Example 1: Sign of the base

Evaluate (−2)⁵, −2⁴, (−1)¹⁰⁰, −5⁰.

  1. (−2)⁵: odd exponent, negative base, so negative. 2⁵ = 32, so (−2)⁵ = −32.
  2. −2⁴: the base is 2. 2⁴ = 16, then apply the negative: −16.
  3. (−1)¹⁰⁰: even exponent, so positive: 1.
  4. −5⁰: the base is 5, and 5⁰ = 1, so −5⁰ = −1. (Compare (−5)⁰ = 1.)

−32, −16, 1, −1

Worked example

Example 2: Negative exponents

Evaluate 4⁻², (1/3)⁻², (−2)⁻³ and 0.5⁻¹.

  1. 4⁻² = 1/4² = 1/16.
  2. (1/3)⁻² = (3/1)² = 3² = 9. A negative exponent flips the fraction.
  3. (−2)⁻³ = 1/(−2)³ = 1/(−8) = −1/8. The base is negative and the exponent is odd, so the value is negative.
  4. 0.5⁻¹ = 1/0.5 = 2.

1/16, 9, −1/8, 2

Worked example

Example 3: Put powers in order

Order from least to greatest: 2⁻³, (1/2)², 3⁰, (−2)³, 2³

  1. Evaluate each: 2⁻³ = 1/8 = 0.125; (1/2)² = 1/4 = 0.25; 3⁰ = 1; (−2)³ = −8; 2³ = 8.
  2. Order the values: −8 < 0.125 < 0.25 < 1 < 8.

(−2)³, 2⁻³, (1/2)², 3⁰, 2³

Worked example

Example 4: Powers in the order of operations

Evaluate 3² × 2⁻² + (−1)³.

  1. Powers first: 3² = 9, 2⁻² = 1/4, (−1)³ = −1.
  2. Multiply: 9 × 1/4 = 9/4.
  3. Add: 9/4 + (−1) = 9/4 − 4/4 = 5/4.

5/4 (or 1.25)

Hands-on activity

Folding powers

You need: a sheet of paper, a pencil.

  1. Fold the sheet in half. Count the layers (2). Fold again (4 layers). Keep going as long as you can.
  2. Make a table: number of folds n, number of layers, and the fraction of the original sheet's area that the top face covers.
  3. Find the pattern. Layers = 2ⁿ. Area of the top face = (1/2)ⁿ = 2⁻ⁿ of the sheet.
  4. Unfold the sheet back one step at a time and say each power out loud: 2³ = 8 layers, 2² = 4, 2¹ = 2, 2⁰ = 1 (no folds, one layer!).

Model answer after 5 folds: 2⁵ = 32 layers, top face = 2⁻⁵ = 1/32 of the sheet.

Question: Why does it get so hard to fold after 6 or 7 folds? (The stack is 2⁶ = 64 or 2⁷ = 128 sheets thick.)

Just for fun

A funny-but-true fact: people used to say a sheet of paper can't be folded in half more than 7 times. In 2002, high-school student Britney Gallivan folded a very long strip of paper in half 12 times. That's 2¹² = 4096 layers!

Your turn

1

Name the base and exponent in (−4)³, then evaluate it.

2

Evaluate −4² and (−4)².

3

Evaluate 7⁰ and −7⁰.

4

Evaluate 5⁻².

5

Evaluate (2/5)⁻².

6

Evaluate (−3)⁻³.

7

Which is greater, (0.9)² or (0.9)³? Explain without calculating.

8

Evaluate 2⁻¹ + 2⁰ + 2¹.

More practice: warm-up

1

Evaluate 2⁶.

2

Evaluate 10⁻².

3

Evaluate (−1)⁷.

4

Evaluate 9⁰.

5

Evaluate (1/2)³.

More practice: on your own

1

Order from least to greatest: 3⁻¹, (−3)², −3², 3⁰.

2

Evaluate (−2)³ − 2⁻².

3

Evaluate (3/4)⁻¹ × 6⁰ × 2².

4

A colony of 800 bacteria doubles every hour. Use 800 × 2ⁿ to find the number 3 hours before (n = −3).

5

Without calculating, is (−5)¹⁵ positive or negative? Is (−5)⁻¹⁴ positive or negative?

Challenge

1

Use the pattern "divide by the base each step" to explain why a⁰ = 1 for any a ≠ 0.

2

Find the value of n: 2ⁿ = 1/32.

3

For which values of a is a² < a? Explain with examples.

Check your understanding

Quiz

1.

What is (−3)⁴?

2.

What is −3⁴?

3.

What is 6⁻²?

4.

What is (2/3)⁻²?

5.

Which value is the smallest?

6.

Which is equal to 1?

For parents and teachers

Big idea: Expectation B2.1 asks students to analyse, through patterns, how the sign and size of an exponent affect the value of a power. The halving pattern makes a⁰ = 1 and a⁻ⁿ = 1/aⁿ make sense instead of being rules to memorize.

Common difficulties:

  • Thinking a negative exponent makes a negative number (writing 2⁻³ = −8). It means reciprocal: 1/8.
  • Multiplying base by exponent (2⁵ = 10). Ask "how many 2s are multiplied?"
  • Ignoring brackets: −3² = −9 but (−3)² = 9. Many calculators also follow this, so typing −3² gives −9.
  • Thinking anything to the 0 is 0. It is 1 (for a non-zero base).

How to help: Do the folding activity together. Ask your teen to predict the sign of (−1)²⁰²⁶ and (−1)²⁰²⁷ and explain.