Chapter 2 of 8

All 8 chapters
  1. 1Laws of powers, readunderstood
  2. 2Zero and negative powers, readunderstood
  3. 3Roots and fractional powers, readunderstood
  4. 4Standard form, readunderstood
  5. 5Calculating in standard form, readunderstood
  6. 6Rational and irrational numbers, readunderstood
  7. 7Surds, readunderstood
  8. 8Accuracy and bounds, readunderstood

M4.1.2 Chapter 2

Zero and negative powers

You cannot multiply by 5 no times, so what can 5⁰ mean?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Explain why 5⁰ = 1
  • Write 2⁻³ as 1/8

Mei writes the powers of 2 in a column: 2³ = 8, then 2² = 4, then 2¹ = 2. She sees a pattern. Each step down divides the answer by 2.

Draw it first

Keep going down the ladder

  1. A ladder of powers of 2 with a down arrow labelled ÷ 2 between each pair of rows. Step 1: 2³ = 8, 2² = 4 and 2¹ = 2. Step 2: the row 2⁰ = 1, in a box. Step 3: the rows 2⁻¹ = 1/2, 2⁻² = 1/4 and 2⁻³ = 1/8.2³=8÷ 22²=4÷ 22¹=2

    1 Start with 2³ = 8, 2² = 4 and 2¹ = 2. Each step down divides by 2.

  2. A ladder of powers of 2 with a down arrow labelled ÷ 2 between each pair of rows. Step 1: 2³ = 8, 2² = 4 and 2¹ = 2. Step 2: the row 2⁰ = 1, in a box. Step 3: the rows 2⁻¹ = 1/2, 2⁻² = 1/4 and 2⁻³ = 1/8.2³=8÷ 22²=4÷ 22¹=2÷ 22⁰=1

    2 Divide by 2 again. 2 ÷ 2 = 1, so 2⁰ = 1.

  3. A ladder of powers of 2 with a down arrow labelled ÷ 2 between each pair of rows. Step 1: 2³ = 8, 2² = 4 and 2¹ = 2. Step 2: the row 2⁰ = 1, in a box. Step 3: the rows 2⁻¹ = 1/2, 2⁻² = 1/4 and 2⁻³ = 1/8.2³=8÷ 22²=4÷ 22¹=2÷ 22⁰=1÷ 22⁻¹=1/2÷ 22⁻²=1/4÷ 22⁻³=1/8

    3 Keep dividing by 2 below zero: 2⁻¹ = 1/2, 2⁻² = 1/4 and 2⁻³ = 1/8.

The same thing happens for any number a except 0. a⁰ = 1. We leave 0⁰ alone, because it does not fit the pattern.

A ladder of powers of 5 with a down arrow labelled ÷ 5 between each pair of rows: 5² = 25, 5¹ = 5 and 5⁰ = 1 in a box.5²=25÷ 55¹=5÷ 55⁰=1
The same ladder for 5. One step below 5¹ = 5 is 5 ÷ 5 = 1.

The second law gives the same answer. Take 5³ ÷ 5³. Subtracting the indices gives 3 − 3 = 0, so it is 5⁰. But any number except 0, divided by itself, is 1. So 5⁰ = 1.

Worked example

Work out 7⁰.

  1. 7¹ ÷ 7¹ has index 1 − 1 = 0, so it equals 7⁰.
  2. 7 ÷ 7 = 1.
  3. So 7⁰ = 1.

Answer: 7⁰ = 1

Now go below zero. The ladder carries on dividing by 2, so 2⁻¹ = 1/2. A negative index means one over the positive power. 2⁻³ = 1/2³. The reciprocal of a number is 1 divided by it.

A ladder of powers of 10 with a down arrow labelled ÷ 10 between each pair of rows: 10³ = 1 000, 10² = 100, 10¹ = 10, 10⁰ = 1 in a box, then 10⁻¹ = 1/10, 10⁻² = 1/100 and 10⁻³ = 1/1 000.10³=1 000÷ 1010²=100÷ 1010¹=10÷ 1010⁰=1÷ 1010⁻¹=1/10÷ 1010⁻²=1/100÷ 1010⁻³=1/1 000
Powers of 10 going down. 10⁻³ = 1/1 000, which is 0.001 as a decimal.

Worked example

Write 2⁻³ as a fraction.

  1. A negative index means 1 divided by the positive power.
  2. 2³ means 2 × 2 × 2 = 8.
  3. So 2⁻³ = 1/8.

Answer: 2⁻³ = 1/8

Worked example

Write 5⁴ ÷ 5⁶ as a fraction.

  1. Subtract the indices: 4 − 6 = −2, so it is 5⁻².
  2. A negative index means one over: 5⁻² = 1/5².
  3. 5² = 25.
  4. So 5⁴ ÷ 5⁶ = 1/25.

Answer: 5⁴ ÷ 5⁶ = 1/25

See it three ways

With things

Fold a long strip of paper in half. Fold it in half again, and once more. Open it out.

As a picture

A strip cut into 8 equal parts with one part shaded. The label under it says 2⁻³ of the strip.2⁻³ of the strip

In symbols

Each fold halves the part: 1/2 × 1/2 × 1/2 = 1/8. That is 2⁻³ = 1/8.

Try it

Fold and count

You need: a long strip of paper, a pencil.

  1. Fold the strip in half and open it. Count the parts.
  2. Fold it in half twice and open it. Count again.
  3. Fold it in half three times and open it. Count again.

What you will see: One fold gives 2 equal parts, two folds give 4, and three folds give 8. One part is 1/2, then 1/4, then 1/8 of the strip.

If you cannot try it, this is what happens.
A table with the head Folds, Parts and One part. Rows: 0, 1, 1; 1, 2, 1/2; 2, 4, 1/4; 3, 8, 1/8.FoldsPartsOne part011121/2241/4381/8

A wrong idea many people have

2⁻³ = −8, because of the minus sign.

Why it feels right: A minus sign usually makes a number negative.

The test that breaks it: Follow the ladder: 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2. The numbers get smaller, but they stay above 0. They never reach −8.

What is true: A negative index means one over the positive power. 2⁻³ = 1/8, a small positive number.

A wrong idea many people have

5⁰ = 0, because there are no 5s.

Why it feels right: Zero often means nothing.

The test that breaks it: On the ladder 5² = 25 and 5¹ = 5, each step divides by 5. The next step is 5 ÷ 5 = 1, not 0.

What is true: a⁰ = 1 for every number a except 0.

Key words

reciprocal
one divided by a number. The reciprocal of 8 is 1/8.

Back to your guess

The answer is B: 5⁰ = 1, and a pattern of dividing by 5 shows why.

Each step down the powers divides by 5, and 5 ÷ 5 = 1. So 5⁰ = 1. It is not 0 and it is not 5. The same pattern works for any number except 0.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is 9⁰?

    Tap your answer.

    1. Yes, that is right. Any number except 0 to the power 0 is 1. Here 9 ÷ 9 = 1.

    2. Not this one. You may think no copies of 9 means nothing. But each step down the ladder divides by 9, and 9 ÷ 9 is 1. Go to M4.1.2. Try another.

    3. Not this one. The index 0 does not leave the 9 as it is. That would be the index 1. On the ladder 9¹ = 9, and the next step down divides by 9. Go to M4.1.2. Try another.

    See the answer

    1

    Any number except 0 to the power 0 is 1. Here 9 ÷ 9 = 1.

    • If you chose B: You may think no copies of 9 means nothing. But each step down the ladder divides by 9, and 9 ÷ 9 is 1. Go to M4.1.2.
    • If you chose C: The index 0 does not leave the 9 as it is. That would be the index 1. On the ladder 9¹ = 9, and the next step down divides by 9. Go to M4.1.2.

  2. The ladder shows powers of 3. What is 3⁻²?

    A ladder of powers of 3 with a down arrow labelled ÷ 3 between each pair of rows: 3² = 9, 3¹ = 3, 3⁰ = 1 in a box and 3⁻¹ = 1/3.3²=9÷ 33¹=3÷ 33⁰=1÷ 33⁻¹=1/3
    Tap your answer.

    1. Not this one. The minus sign in the index does not make the answer negative. The ladder stays above 0 as it goes down. Go to M4.1.2. Try another.

    2. Not this one. You worked out 3 × 2 = 6. The index 2 means two 3s multiplied: 3 × 3 = 9. Then take one over that. Go to M4.1.2. Try another.

    3. Not this one. A negative index flips the number upside down. 9 is the positive power, 3². The answer is one over that. Go to M4.1.2. Try another.

    4. Yes, that is right. Divide 1/3 by 3 once more: it is 1/9. Or use the rule: 3⁻² = 1/3² = 1/9.

    See the answer

    1/9

    Divide 1/3 by 3 once more: it is 1/9. Or use the rule: 3⁻² = 1/3² = 1/9.

    • If you chose A: The minus sign in the index does not make the answer negative. The ladder stays above 0 as it goes down. Go to M4.1.2.
    • If you chose B: You worked out 3 × 2 = 6. The index 2 means two 3s multiplied: 3 × 3 = 9. Then take one over that. Go to M4.1.2.
    • If you chose C: A negative index flips the number upside down. 9 is the positive power, 3². The answer is one over that. Go to M4.1.2.

  3. Which number is the same as 10⁻³?

    Tap your answer.

    1. Not this one. You put one zero too many. 10⁻³ = 1/1 000, and the 1 sits in the third place after the point. Go to M4.1.2. Try another.

    2. Not this one. A negative index does not make the number negative or big. It makes it small: 1/1 000. Go to M4.1.2. Try another.

    3. Yes, that is right. 10⁻³ = 1/10³ = 1/1 000. One thousandth is 0.001.

    See the answer

    0.001

    10⁻³ = 1/10³ = 1/1 000. One thousandth is 0.001.

    • If you chose A: You put one zero too many. 10⁻³ = 1/1 000, and the 1 sits in the third place after the point. Go to M4.1.2.
    • If you chose B: A negative index does not make the number negative or big. It makes it small: 1/1 000. Go to M4.1.2.

  4. Which sentence explains why 5⁰ = 1?

    Tap your answer.

    1. Not this one. No copies is not the same as the answer 0. Going down the ladder divides by 5 each time, and that reaches 1. Go to M4.1.2. Try another.

    2. Yes, that is right. 5¹ ÷ 5¹ has index 1 − 1 = 0, and 5 ÷ 5 = 1.

    3. Not this one. The index 0 is a real step on the ladder. 5¹ = 5, and one step down is 5 ÷ 5 = 1. Go to M4.1.2. Try another.

    See the answer

    The second law gives a number divided by itself the index zero, and any number except zero, divided by itself, is one.

    5¹ ÷ 5¹ has index 1 − 1 = 0, and 5 ÷ 5 = 1.

    • If you chose A: No copies is not the same as the answer 0. Going down the ladder divides by 5 each time, and that reaches 1. Go to M4.1.2.
    • If you chose C: The index 0 is a real step on the ladder. 5¹ = 5, and one step down is 5 ÷ 5 = 1. Go to M4.1.2.

  5. Which is the same as 3⁵ ÷ 3⁷?

    Tap your answer.

    1. Yes, that is right. 5 − 7 = −2, so the power is 3⁻². And 3⁻² = 1/3² = 1/9.

    2. Not this one. You took 7 − 5 instead of 5 − 7. The top index is smaller, so the answer is below 1: one over 9. Go to M4.1.2. Try another.

    3. Not this one. You added the indices. Dividing subtracts them: 5 − 7 = −2, so the power is 3⁻², which is 1/9. Go to M4.1.1. Try another.

    See the answer

    1/9

    5 − 7 = −2, so the power is 3⁻². And 3⁻² = 1/3² = 1/9.

    • If you chose B: You took 7 − 5 instead of 5 − 7. The top index is smaller, so the answer is below 1: one over 9. Go to M4.1.2.
    • If you chose C: You added the indices. Dividing subtracts them: 5 − 7 = −2, so the power is 3⁻², which is 1/9. Go to M4.1.1.

  6. From an earlier chapter

    Simplify a⁹ ÷ a⁴. What is the index in the answer?

    See the answer

    5

    Subtract the indices: 9 − 4 = 5. So the answer is a⁵.

  7. Write 2⁻³ as a decimal.

    See the answer

    0.125

    2⁻³ = 1/2³ = 1/8. And 1 ÷ 8 = 0.125.

  8. Work out 4⁰ + 2⁻¹. Give the answer as a decimal.

    See the answer

    1.5

    4⁰ = 1 and 2⁻¹ = 1/2 = 0.5. Together: 1 + 0.5 = 1.5.

  9. Which whole number n makes 1/32 equal to 2 to the power −n?

    See the answer

    5

    32 = 2 × 2 × 2 × 2 × 2 = 2⁵. So 1/32 = 2⁻⁵, and n = 5.

Keep it

Copy this into your exercise book.

a⁰ = 1 for any a except 0. A negative index means one over: 2⁻³ = 1/2³ = 1/8. Going down the ladder divides each time. The answer stays above 0.

Draw: A ladder of powers of 2, from 2³ = 8 to 2⁻³ = 1/8, with a ÷ 2 arrow between each pair of rows.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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