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

Front numbers together, powers of ten together

To multiply or divide, work the front numbers and the powers of ten separately, then check the front number is still below 10. To add, first make the powers of ten match.

Number and AlgebraScientific calculator

Warm-up

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

  1. (Lesson 6) Write 0.0049 in scientific notation.
    Answer4.9 × 10−3, because the point moves 3 places right.
  2. (Lesson 5) Simplify 105 × 10−2.
    Answer103, because 5 + (−2) = 3.
  3. (Lesson 6) Which is greater, 8 × 10−4 or 2 × 10−3?
    Answer2 × 10−3, because the exponent −3 is greater than −4.
Words used in this lesson
front number
the number from 1 up to 10 in scientific notation. For example: 2.5 in 2.5 × 106.
carry
move a factor of ten into the exponent. For example: 10 × 108 = 1 × 109.
EXP key
the calculator key for “times ten to the power”. For example: 2.5 EXP 6.
estimate
a rough answer worked out first. For example: 10 × 108.
significant figures
the digits that carry accuracy. For example: 2.5 has 2 significant figures.
match
write with the same power of ten. For example: 3.2 × 104 = 0.032 × 106.

Multiplying and dividing

A number in scientific notation is a front number times a power of ten, and multiplication lets you regroup: (2 × 103) × (3 × 104) = (2 × 3) × (103 × 104) = 6 × 107. The front numbers multiply and the powers of ten follow the product law from Lesson 3. Predict first: what is (3 × 108) × (4 × 10−3)? Then try the figure.

2.5
6
4
−3
10front numbers multiplied
3exponents added
1 × 10⁴in scientific notation

(2.5 × 10⁶) × (4 × 10⁻³): 2.5 × 4 = 10 and 6 + (−3) = 3, so 10 × 10³ = 1 × 10⁴.

Show me the steps
  1. (2.5 × 10⁶) × (4 × 10⁻³): 2.5 × 4 = 10 and 6 + (−3) = 3, so 10 × 10³ = 1 × 10⁴.
  2. (4.2 × 10³) × (3 × 10⁴): 4.2 × 3 = 12.6 and 3 + 4 = 7, so 12.6 × 10⁷ = 1.26 × 10⁸.
  3. (1.5 × 10⁻⁴) × (2 × 10⁻²): 1.5 × 2 = 3 and −4 + (−2) = −6, so 3 × 10⁻⁶. No carry is needed.
  4. (8 × 10⁷) × (9 × 10⁻⁷): 8 × 9 = 72 and 7 + (−7) = 0, so 72 × 10⁰ = 7.2 × 10¹.
  5. (3.1 × 10⁻³) × (3.1 × 10⁻³): 3.1 × 3.1 = 9.61 and −3 + (−3) = −6, so 9.61 × 10⁻⁶. No carry is needed.

For the prediction: 3 × 4 = 12 and 8 + (−3) = 5, so the product is 12 × 105. The front number 12 is not below 10, so write 12 = 1.2 × 10, and the answer is 1.2 × 106.

To multiply: multiply the front numbers, add the exponents. To divide: divide the front numbers, subtract the exponents. Then check that the front number is still at least 1 and less than 10; if it is not, move one factor of ten into the exponent. To add or subtract: first rewrite so both have the same power of ten, then add or subtract the front numbers and leave the power as it is.

An answer cannot be more accurate than the data. If the data has two significant figures, give the answer to two or three, not all the digits a calculator shows.

Adding

Adding is different. You cannot add 4.5 × 106 and 3.2 × 104 by adding the front numbers and the exponents, because they count different sizes of ten. First write 3.2 × 104 as 0.032 × 106, and then the front numbers add.

4.5
3.2
6
2
0.032second, as a front number of 10⁶
4.532front numbers added
4.532 × 10⁶in scientific notation

4.5 × 10⁶ + 3.2 × 10⁴ = 4.5 × 10⁶ + 0.032 × 10⁶ = 4.532 × 10⁶.

Show me the steps
  1. 4.5 × 10⁶ + 3.2 × 10⁴ = 4.5 × 10⁶ + 0.032 × 10⁶ = 4.532 × 10⁶.
  2. 9.6 × 10⁵ + 7.5 × 10⁵: the powers already match, 9.6 + 7.5 = 17.1, so 17.1 × 10⁵ = 1.71 × 10⁶.
  3. 2 × 10⁸ + 5 × 10⁵ = 2 × 10⁸ + 0.005 × 10⁸ = 2.005 × 10⁸.
  4. 6.8 × 10⁷ + 9.9 × 10⁶ = 6.8 × 10⁷ + 0.99 × 10⁷ = 7.79 × 10⁷.

A common mistake: adding the exponents when adding the numbers. (3 × 104) + (2 × 104) = 5 × 104, not 5 × 108. Exponents add when numbers are multiplied, because multiplying powers of ten joins runs of tens. When numbers are added, the power of ten is the unit, and units stay the same.

Worked example: a carry

Calculate (4 × 105) × (7 × 103).

  1. Front numbers: 4 × 7 = 28. Powers of ten: 105 × 103 = 108.
  2. So the product is 28 × 108, and 28 is not below 10.
  3. 28 = 2.8 × 10, so 28 × 108 = 2.8 × 109.

Worked example: dividing

Calculate (9.6 × 107) ÷ (3 × 10−2).

  1. Front numbers: 9.6 ÷ 3 = 3.2.
  2. Powers of ten: 7 − (−2) = 9.
  3. So the answer is 3.2 × 109, with the front number already below 10.

Your go. The first step is given. Calculate (6 × 10−4) × (5 × 107).

  1. Front numbers: 6 × 5 = 30. Powers of ten: −4 + 7 = 3.
  2. Now fix the front number, then check.
Check the answer

30 × 103 = 3 × 10 × 103 = 3 × 104.

With a calculator

A scientific calculator has a key for “times ten to the power”, usually marked EXP or ×10x. To enter 2.5 × 106, press 2.5, then the key, then 6. The display shows something like 2.5E6 (calculator notation); the E is the same “times 10 to the”. Do not also type × 10 yourself.

Estimate before you press anything. (3.5 × 105) × (2 × 103) is about 3.5 × 2 = 7, times 108, so about 7 × 108. If the calculator shows 7E8, it agrees. If it shows 7E7 or 7E9, a key press has slipped. The estimate catches it; the calculator does not.

Check before you go on

1. What is (3 × 104) × (2 × 105)?

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. (2.5 × 105) × (8 × 103) = a × 10n with 1 ≤ a < 10. What is n?

3. (8 × 109) ÷ (2 × 103) = a × 10n with 1 ≤ a < 10. What is n?

Level 2 · Building

4. 6 × 105 + 8 × 104 = a × 105. What is a?

5. (1.2 × 10−3) × (5 × 10−2) = a × 10n with 1 ≤ a < 10. What is n?

6. 5 × 106 − 4 × 105 = a × 106. What is a?

Level 3 · Stretch

7. Light travels at 3.0 × 108 metres each second and takes 4.8 × 102 seconds to come from the Sun. Estimate first, then use a calculator to check. The distance is a × 1011 metres. What is a to two significant figures?

8. A transistor is 2 × 10−8 m wide. How many fit side by side across a gap of 1 × 10−3 m? Give the answer as an ordinary number.

Something to think about

The Sun is about 1.5 × 108 km from the Earth, and the Moon about 3.8 × 105 km. About how many times further away is the Sun? Estimate before you divide.

One way to think about it

1.5 ÷ 3.8 is about 0.4, and 108 ÷ 105 = 103, so the Sun is about 400 times as far. Dividing exactly gives 394.7…, and the data has two significant figures, so the answer is about 390 times.

Exit ticket

9. Ife adds 3 × 104 and 5 × 103 and writes 8 × 107. What went wrong, and what is the right answer?