Year 9 · Unit 1 Real numbers and indices · Lesson 5 of 7

What lies below the first power?

Walk a ladder of powers downwards, dividing by the base at every step. The pattern does not stop at the first power: it passes through 1 and into fractions, and the laws of indices still hold.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 3) Simplify 79 ÷ 74.
    Answer75, because the exponents subtract: 9 − 4 = 5.
  2. (Lesson 4) Write (23)4 as one power.
    Answer212, because the exponents multiply: 3 × 4 = 12.
  3. (Year 7, Unit 5, lesson 6) What is 1 ÷ (1/4)?
    Answer4, because dividing by 1/4 is the same as multiplying by 4.
Words used in this lesson
zero exponent
an exponent of 0, which gives 1. For example: 60 = 1.
negative exponent
an exponent below 0, which gives a reciprocal. For example: 2−3 = 1/8.
reciprocal
one divided by a number. For example: the reciprocal of 4 is 1/4.
positive exponent
an exponent above 0. For example: 3 in 53.
flip
swap the top and the bottom of a fraction. For example: 2/3 flips to 3/2.
simplify
write in a shorter or simpler form. For example: 95 × 9−8 = 1/729.

Keep dividing

Powers of 2 count upwards: 23 = 8, 22 = 4, 21 = 2. Look at the pattern going down. Each step takes one factor of 2 away, so each value is half of the one before. What does the pattern say comes next? Predict first: what are 100 and 10−1? Then set the base and the exponent in the ladder and read the rungs.

3
2
3²power
9value

3² = 9: 2 factors of 3.

Show me the steps
  1. 2³ = 8, 2² = 4, 2¹ = 2: each step down divides by 2, so the next rung is 2⁰ = 1.
  2. 2⁰ = 1: dividing 2 by 2 gives 1, so the pattern puts 1 here.
  3. 2⁻³ = 1/8: from 1, divide by 2 three more times: 1/2, 1/4, 1/8.
  4. 10⁻² = 1/100 = 0.01: from 1, divide by 10 twice.
  5. 3⁻² = 1/9: from 1, divide by 3 twice.
  6. 5⁰ = 1: the same 1 for every base, because the step down from the base is always a division by itself.

The pattern passes through 1 at exponent 0 and goes on into fractions, each one a division by the base further down. That is a definition, but it is not arbitrary: it is the only one that keeps the laws of indices true. Take the quotient law with equal exponents: 23 ÷ 23 = 23 − 3 = 20. But 8 ÷ 8 = 1, so 20 has to be 1. In the same way, 22 ÷ 25 = 2−3, and cancelling gives (2 × 2) / (2 × 2 × 2 × 2 × 2) = 1/(2 × 2 × 2), so 2−3 = 1/8. For the prediction, the same steps give 100 = 1 and 10−1 = 1/10 = 0.1.

For any number a that is not 0: a0 = 1 and a−n = 1/an. A negative exponent does not make the number negative. It says how many times to divide 1 by the base. The laws of indices work with zero and negative exponents too, so the answer to a question can always be written with positive exponents.

For a fraction, a negative exponent flips it: (2/3)−2 = (3/2)2 = 9/4.

A common mistake: “2−3 = −8.” The minus sign is in the exponent, not in the number. 2−3 = 1/8, which is small and positive. A power of a positive number is never negative, and never reaches 0.

Worked example: a negative exponent as a fraction

Write 5−3 as a fraction and as a decimal.

  1. A negative exponent means one over the power: 5−3 = 1/53.
  2. 53 = 125, so the fraction is 1/125.
  3. 1 ÷ 125 = 0.008. Check: 125 × 0.008 = 1.

Worked example: the laws with negative exponents

Simplify 95 × 9−8, giving the answer with a positive exponent.

  1. Same base, so add the exponents: 5 + (−8) = −3.
  2. 9−3 = 1/93.
  3. 93 = 729, so the answer is 1/729.

Your go. The first step is given. Write 73 × 7−5 as a fraction.

  1. Add the exponents: 3 + (−5) = −2.
  2. Write the negative power as a fraction, then check.
Check the answer

7−2 = 1/72 = 1/49.

Check before you go on

1. What is 60?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Write 10−3 as a decimal.

3. What is 80 + 50?

Level 2 · Building

4. 23 × 2−9 = 1/d. What is d?

5. Evaluate (1/2)−3.

6. Which list shows 2−1, 20, 2−3 and 21 in order from smallest to greatest?

Level 3 · Stretch

7. A dose of medicine halves in the body every 6 hours. After 30 hours the fraction left is 2k. What is k?

8. Write 2−1 + 2−2 as a decimal.

Something to think about

A bacteria colony doubles every hour. There are 1000 now. How many were there 3 hours ago, written with a power of 2? What does a negative exponent mean here?

One way to think about it

Each hour back divides by 2, so 3 hours ago there were 1000 ÷ 2 ÷ 2 ÷ 2 = 125, which is 1000 × 2−3. A negative exponent counts steps backwards in time from the starting value 20 = 1.

Exit ticket

9. Khalid says 4−2 = −8, because 4 × (−2) = −8. What should he do?