Year 9 · Unit 1 Real numbers and indices · Lesson 3 of 7

Same base: add to multiply, subtract to divide

Write a power out as repeated multiplication and the first two laws of indices appear: joining two runs of the same factor adds the exponents, and cancelling takes them away.

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. (Year 7, Unit 1, lesson 2) What is 103?
    Answer1000, because 10 × 10 × 10 = 1000.
  2. (Year 8, Unit 3, lesson 5) What is (−2)3?
    Answer−8, because (−2) × (−2) × (−2) = −8.
  3. (Lesson 2) What fraction is 0.555…?
    Answer5/9, because 9x = 5.
Words used in this lesson
power
a number multiplied by itself a number of times. For example: 23 = 2 × 2 × 2.
base
the number that is multiplied by itself. For example: 2 in 23.
exponent
the number of factors of the base; also called the index. For example: 3 in 23.
factor
a number that is multiplied. For example: each 2 in 2 × 2 × 2.
coefficient
the number in front of a letter or power. For example: 15 in 15a6.
indices
more than one index, or the laws about exponents. For example: the laws of indices.

Powers as runs of factors

A power is repeated multiplication. In 23 the base is 2 and the exponent is 3: it says 3 factors of 2, so 23 = 2 × 2 × 2 = 8. A folder holds 23 files and each file is 24 bytes. How many bytes is the folder, written as one power of 2? Predict first: is it 27, 212 or 47? Then use the figure, and try a different base.

5
2
3
5⁵as one power
3125value
2 + 3 = 5exponents

5² × 5³ = 5⁵ = 3125: 2 + 3 = 5 factors of 5 in a row.

Show me the steps
  1. 2³ × 2⁴ = 2⁷ = 128: 3 + 4 = 7 factors of 2 in a row.
  2. 3² × 3⁵ = 3⁷ = 2187: 2 + 5 = 7 factors of 3.
  3. 10⁴ × 10² = 10⁶ = 1000000: 4 + 2 = 6 factors of 10.
  4. 7¹ × 7³ = 7⁴ = 2401: a number with no written exponent has exponent 1, so 7 = 7¹ and 1 + 3 = 4.
  5. 5⁶ ÷ 5² = 5⁴ = 625: 2 of the 6 factors cancel, and 6 − 2 = 4 are left.
  6. 6⁵ ÷ 6³ = 6² = 36: 3 of the 5 factors cancel, and 5 − 3 = 2 are left.

Written out, the idea is plain. 23 × 24 = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 27, which is 128. Seven factors of 2 in a row is 3 + 4. Dividing cancels factors from the top and bottom: 27 ÷ 23 leaves 7 − 3 = 4 factors, so it is 24 = 16.

For the same base: am × an = am + n, and am ÷ an = am − n. The base stays the same; only the exponents add or subtract. A base with no written exponent has the exponent 1: a = a1.

With numbers in front, multiply or divide those numbers separately from the powers: 3a4 × 5a2 = 15a6.

A common mistake: multiplying the bases. 23 × 24 is not 47. The factor is still 2; there are 7 of them. Check: 8 × 16 = 128 = 27, but 47 = 16384. The law also needs the same base: 23 × 34 does not combine into one power with the laws above.

Worked example: one power from a quotient

Write 48 ÷ 45 as one power, then find its value.

  1. The base is 4 in both, so subtract the exponents: 8 − 5 = 3.
  2. 48 ÷ 45 = 43.
  3. 4 × 4 × 4 = 64.

Worked example: with numbers in front

Simplify 3a4 × 5a2.

  1. Numbers: 3 × 5 = 15.
  2. Powers of a: a4 × a2 = a4 + 2 = a6.
  3. Together: 15a6.

Your go. The first step is given. Simplify 20m7 ÷ 5m2.

  1. Numbers: 20 ÷ 5 = 4.
  2. Now do the powers of m, then put them together and check.
Check the answer

m7 ÷ m2 = m7 − 2 = m5, so the answer is 4m5.

Check before you go on

1. Which is 54 × 53 written as a single power?

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. 36 × 3k = 310. What is k?

3. Evaluate 29 ÷ 26.

Level 2 · Building

4. 4p3 × 5p8 = 20pk. What is k?

5. 18c9 ÷ 6c4 is a number times c5. What is the number?

6. Which of these is NOT equal to 210?

Level 3 · Stretch

7. (79 × 74) ÷ 710 = 7k. What is k?

8. A phone's drive holds 230 bytes and one photo takes 220 bytes. How many photos fit?

Something to think about

Is 23 + 24 equal to 27? Test it by working out both sides.

One way to think about it

No: 8 + 16 = 24 but 27 = 128. The law joins runs of factors when numbers are multiplied; adding does not join anything.

Exit ticket

9. Ravi says: “9 × 35 cannot be written as one power of 3, because the bases 9 and 3 are different.” What is the best reply?