Year 10 Extended · Unit 1 Number sets, surds, indices and bounds · Lesson 2 of 6

How do powers of ten do the work in a unit conversion?

Very large and very small quantities, and units built from two units such as g/cm³, all run on powers of ten. This lesson asks what happens to the exponents when you multiply, divide, add and convert.

Number and AlgebraGDC optional

Warm-up

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

  1. From Year 9: write 0.00072 in standard form.
    Answer7.2 × 10⁻⁴. The first non-zero digit is 4 places after the point.
  2. Work out 10³ × 10⁻⁷.
    Answer10⁻⁴. Add the exponents: 3 + (−7) = −4.
  3. From Year 9: how many mm² are there in 1 cm²?
    Answer100. 1 cm = 10 mm, and area is length times length.
Words used in this lesson
standard form
a number written as a × 10ⁿ with 1 ≤ a < 10 and n a whole number. For example: 4.2 × 10⁻³.
coefficient
the number between 1 and 10 in front of the power of ten. For example: the 6.5 in 6.5 × 10⁸.
compound unit
a unit built from two units, one divided by the other. For example: metres per second.
density
mass divided by volume. For example: the mass in each cm³.
conversion factor
a ratio equal to 1 that changes the unit but not the amount. For example: 1 m / 100 cm.
carry
when the coefficient reaches 10 or more and the exponent goes up by 1. For example: 40 × 10² becomes 4 × 10³.

The Sun in hydrogen atoms

The Sun has a mass of about 2 × 10³⁰ kg, and a hydrogen atom has a mass of about 1.7 × 10⁻²⁷ kg. Dividing one by the other tells you roughly how many hydrogen atoms the Sun is made of. Nobody would do that on paper in ordinary decimal form: the numbers have 30 and 27 zeros.

Predict first: will the exponent of the answer be 3, 57 or 30 ÷ 27? And if you convert 1 m³ to cm³, will the number go up by a factor of 100, 1000 or 1 000 000? Use the figures, then check.

2
2 × 10³ kg/m³equals
× 10³net factor

2 g/cm³ = 2 × 10³ kg/m³: g to kg is × 10⁻³ and per cm³ to per m³ is × 10⁶, so the net factor is 10³.

Show me the steps
  1. 2 g/cm³ to kg/m³: grams to kilograms is × 10⁻³, and per cm³ to per m³ is × 10⁶. Net: 10⁻³ × 10⁶ = 10³, so 2 × 10³ kg/m³.
  2. Press per km² to per m²: there are 10⁶ m² in 1 km², so a quantity per km² becomes 10⁶ times smaller per m². One jump, × 10⁻⁶.
  3. Press mg/L to kg/m³: mg to kg is × 10⁻⁶ and per L to per m³ is × 10³ (1 m³ = 1000 L). Net × 10⁻³.
  4. Press kg/m² to g/cm²: kg to g is × 10³ and per m² to per cm² is × 10⁻⁴ (1 m² = 10⁴ cm²). Net × 10⁻¹, so the number gets 10 times smaller.
2
1
3
2
6 × 10³product
no carrycoefficient

2 × 10¹ times 3 × 10² = 6 × 10³, and 6 is already less than 10, so there is no carry.

Show me the steps
  1. 2 × 10¹ times 3 × 10²: the coefficients make 6 and the exponents add to 3. 6 is less than 10, so the answer is 6 × 10³.
  2. Set the coefficients to 4 and 5 with powers 1 and 2: the coefficients make 20, which is 2 × 10, so the answer is 2 × 10⁴. The exponent goes up by 1.
  3. Set both coefficients to 9: 81 = 8.1 × 10, so there is a carry again. On the line, the second bar ends past the next power of ten.
  4. Set one coefficient to 1: the product cannot reach 10, so there is never a carry.

Multiply: (a × 10m)(b × 10n) = ab × 10m + n. Divide: subtract the exponents. If ab is 10 or more, or less than 1, move the point and fix the exponent so that the coefficient is at least 1 and less than 10.

A length factor k becomes k² for area and k³ for volume. A compound unit has two parts, so convert the top and the bottom separately and combine the factors.

To add, first write both with the same power of ten.

Two common mistakes. Adding the exponents when you are adding the numbers: 2 × 10³ + 3 × 10⁴ is not 5 × 10⁷ or 5 × 10³. Write 2 × 10³ = 0.2 × 10⁴, then add: 3.2 × 10⁴. And using the length factor for an area: 1 m = 100 cm, but 1 m² = 10 000 cm², not 100 cm².

Worked examples

1. Work out (3.6 × 10⁻⁴)(5 × 10⁹) in standard form.

  1. Multiply the coefficients: 3.6 × 5 = 18. Add the exponents: −4 + 9 = 5.
  2. 18 × 10⁵ is not in standard form: 18 = 1.8 × 10, so the exponent goes up by 1.
  3. The answer is 1.8 × 10⁶.

2. Divide the Sun's mass by a hydrogen atom's: (2 × 10³⁰) ÷ (1.7 × 10⁻²⁷).

  1. Divide the coefficients: 2 ÷ 1.7 = 1.176…. Subtract the exponents: 30 − (−27) = 57.
  2. 1.176… is between 1 and 10, so no carry is needed.
  3. The Sun is about 1.2 × 10⁵⁷ hydrogen atoms.

3. Iron has a density of 7.9 g/cm³. Give it in kg/m³.

  1. Grams to kilograms: divide by 1000, a factor of 10⁻³.
  2. Per cm³ to per m³: 1 m³ = 100³ cm³ = 10⁶ cm³, so there are 10⁶ times as many cm³ in each m³, a factor of 10⁶.
  3. Net factor 10⁻³ × 10⁶ = 10³: 7.9 × 10³ = 7 900 kg/m³.

Your go. The first step is given. A tank holds 8 × 10⁵ cm³. Write its volume in m³ in standard form.

  1. 1 m = 100 cm, so 1 m³ = 10⁶ cm³. Going from cm³ to m³ divides by 10⁶.
  2. Divide 8 × 10⁵ by 10⁶ and write the answer in standard form.
Check the answer

8 × 10⁵ ÷ 10⁶ = 8 × 10⁻¹, so the tank holds 8 × 10⁻¹ m³ (0.8 m³).

With a GDC (optional)

Your GDC has a standard-form key; its label differs by model, so check the manual. A display such as 2.6E−8 is calculator notation for 2.6 × 10⁻⁸; in written work always write the × 10⁻⁸ form, with a superscript.

Use the GDC to check, not to think: work the exponents by hand first, then compare. When you divide by a number in standard form, put it in brackets: 8 ÷ (2 × 10⁵), because 8 ÷ 2 × 10⁵ is worked out as (8 ÷ 2) × 10⁵. With the exponent key, type 2 EE 5, not 2 × 10 EE 5, which is ten times too big. Look at the answer's size. Everything here can be done by hand.

Check before you go on

1. Sea water has a density of about 1025 kg/m³. What is it in g/cm³?

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 (3.2 × 10⁵) × (4 × 10⁻²) in standard form, as a × 10n. What is n?

3. How many cubic millimetres are there in 5 cm³? Write the answer as 5 × 10n mm³. What is n?

Level 2 · Building

4. A block of aluminium has a volume of 4 m³ and a density of 2.7 g/cm³. Find its mass in kg.

5. A river in Bangladesh carries 5 × 10⁴ m³ of water every second. Written as a × 10n with 1 ≤ a < 10, how many litres per hour is that? What is n? (1 m³ = 1000 L.)

6. Which of these lengths is the greatest? (1 Mm = 10⁶ m.)

Level 3 · Stretch

7. A grain of sand is a cube of side 0.5 mm. Ignoring gaps, about how many grains fill 1 m³? Write the answer as 8 × 10n. What is n?

8. Jupiter's mass is about 1.9 × 10²⁷ kg and Earth's is about 6.0 × 10²⁴ kg. How many times as massive as Earth is Jupiter, to the nearest whole number?

Something to think about

Add 6 × 10⁵ and 7 × 10⁵. A student writes 13 × 10⁵ and stops. What is wrong with that? What should it be? Now add 6 × 10⁵ and 7 × 10³ and decide whether the answer is closer to 6 × 10⁵ or to 13 × 10⁵.

One way to think about it

13 × 10⁵ is a correct value but not in standard form, because 13 is not between 1 and 10: it is 1.3 × 10⁶. For 6 × 10⁵ + 7 × 10³, write 7 × 10³ = 0.07 × 10⁵, so the sum is 6.07 × 10⁵, very close to 6 × 10⁵. Adding needs the same power of ten, and a term two powers smaller barely changes the answer.

Exit ticket

9. Work out (4.0 × 10⁵) + (3.0 × 10⁴) in standard form.