Chapter 5 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

M4.1.5 Chapter 5

Calculating in standard form

How do you multiply 3 × 10⁴ by 2 × 10⁵ without writing out the zeros?

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

  • Multiply and divide numbers in standard form
  • Add numbers in standard form by matching the powers
  • Compare the size of a cell with the Earth in standard form

Hana must work out (3 × 10⁴) × (2 × 10⁵). Writing 30 000 × 200 000 is slow, and one zero too many ruins the answer.

Multiplying can be done in any order. So take the front numbers together, and take the powers of ten together.

Draw it first

Two small jobs instead of one big one

  1. Step 1: the label (3 × 10⁴) × (2 × 10⁵). Step 2: a box 3 × 2 = 6 for the front numbers. Step 3: a box 10⁴ × 10⁵ = 10⁹ for the powers. Step 4: the label = 6 × 10⁹.(3 × 10⁴) × (2 × 10⁵)

    1 Write the two numbers: (3 × 10⁴) × (2 × 10⁵).

  2. Step 1: the label (3 × 10⁴) × (2 × 10⁵). Step 2: a box 3 × 2 = 6 for the front numbers. Step 3: a box 10⁴ × 10⁵ = 10⁹ for the powers. Step 4: the label = 6 × 10⁹.(3 × 10⁴) × (2 × 10⁵)3 × 2 = 6

    2 Multiply the front numbers: 3 × 2 = 6.

  3. Step 1: the label (3 × 10⁴) × (2 × 10⁵). Step 2: a box 3 × 2 = 6 for the front numbers. Step 3: a box 10⁴ × 10⁵ = 10⁹ for the powers. Step 4: the label = 6 × 10⁹.(3 × 10⁴) × (2 × 10⁵)3 × 2 = 610⁴ × 10⁵ = 10⁹

    3 Multiply the powers of ten. The first law adds the indices: 4 + 5 = 9.

  4. Step 1: the label (3 × 10⁴) × (2 × 10⁵). Step 2: a box 3 × 2 = 6 for the front numbers. Step 3: a box 10⁴ × 10⁵ = 10⁹ for the powers. Step 4: the label = 6 × 10⁹.(3 × 10⁴) × (2 × 10⁵)3 × 2 = 610⁴ × 10⁵ = 10⁹= 6 × 10⁹

    4 Put the two answers together: 6 × 10⁹.

Worked example

Work out (3 × 10⁴) × (2 × 10⁵).

  1. Multiply the front numbers: 3 × 2 = 6.
  2. Add the powers of ten: 4 + 5 = 9, because 10⁴ × 10⁵ = 10⁹.
  3. So the answer is 6 × 10⁹.

Answer: 6 × 10⁹

Sometimes the front number comes out as 10 or more. For (4 × 10³) × (5 × 10⁶), the front numbers give 4 × 5 = 20, and the powers give 9. That is 20 × 10⁹, which is not standard form. Write 20 as 2 × 10¹. The answer is 2 × 10¹⁰.

Dividing is the same idea. Divide the front numbers, and subtract the powers: the power on top minus the power below. Then check that the front number is at least 1 and less than 10.

Worked example

Work out (2 × 10⁷) ÷ (8 × 10²).

  1. Divide the front numbers: 2 ÷ 8 = 0.25.
  2. Subtract the powers: 7 − 2 = 5. So the quotient is 0.25 × 10⁵.
  3. 0.25 is below 1, and 0.25 = 2.5 × 10⁻¹.
  4. So 0.25 × 10⁵ = 2.5 × 10⁻¹ × 10⁵ = 2.5 × 10⁴.

Answer: 2.5 × 10⁴

Adding is different. You cannot add 10⁵ and 10⁴ by adding the powers. First change one number so that both have the same power of ten. Then add only the front numbers.

A table with the head Sum, Front and Power. Rows: ×, multiply, add; ÷, divide, subtract; +, add, match first; −, subtract, match first.SumFrontPower×multiplyadd÷dividesubtract+addmatch first−subtractmatch first
What to do to the front numbers (Front) and to the powers of ten (Power).

Worked example

Work out (3.2 × 10⁵) + (4.5 × 10⁴).

  1. The powers are different. Change 4.5 × 10⁴ so that it has the power 5.
  2. 4.5 × 10⁴ is the same as 0.45 × 10⁵. Check: 4.5 ÷ 10 = 0.45.
  3. Now both have 10⁵. Add the front numbers: 3.2 + 0.45 = 3.65.
  4. So the sum is 3.65 × 10⁵, which is 365 000 in full.

Answer: 3.65 × 10⁵

Now a big comparison. A typical animal cell is about 0.00002 m wide, which is 2 × 10⁻⁵ m. The Earth is about 1.3 × 10⁷ m across. The scale shows both, ten times bigger at every tick.

A vertical scale headed Sizes in metres. The ticks, from the bottom to the top, are 10⁻⁶, 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, 1, 10¹, 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷ and 10⁸, each ten times bigger than the one below. Black dots mark cell, ant, adult, Mount Everest and Earth, each at its true height on the scale.Sizes in metres10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹110¹10²10³10⁴10⁵10⁶10⁷10⁸cellantadultMount EverestEarth
Sizes in metres, about: cell 2 × 10⁻⁵, ant 5 × 10⁻³, adult 1.7, Mount Everest 8.8 × 10³, Earth 1.3 × 10⁷.

Worked example

A cell is about 2 × 10⁻⁵ m wide and the Earth is about 1.3 × 10⁷ m across. About how many times wider is the Earth?

  1. Divide the Earth by the cell: (1.3 × 10⁷) ÷ (2 × 10⁻⁵).
  2. Divide the front numbers: 1.3 ÷ 2 = 0.65.
  3. Subtract the powers: 7 − (−5) = 12.
  4. So the quotient is 0.65 × 10¹², and 0.65 = 6.5 × 10⁻¹.
  5. The Earth is about 6.5 × 10¹¹ times wider than a cell.

Answer: About 6.5 × 10¹¹ times wider

See it three ways

With things

Add 3 hundreds and 4 tens. You must change the tens into hundreds first: 4 tens is 0.4 hundreds.

As a picture

A table with the head Written as and Value. Rows: 3 × 10², 300; 0.4 × 10², 40; 3.4 × 10², 340.Written asValue3 × 10²3000.4 × 10²403.4 × 10²340

In symbols

3 × 10² + 4 × 10¹ = 3 × 10² + 0.4 × 10² = 3.4 × 10².

Try it

Check with the full numbers

You need: paper and a pencil.

  1. Write 3 × 10⁴ as 30 000 and 2 × 10⁵ as 200 000.
  2. Multiply the front numbers, then write all the zeros that follow.
  3. Compare with the standard form answer.

What you will see: 3 × 2 = 6, and there are 4 + 5 = 9 zeros. So the product is 6 000 000 000, which is 6 × 10⁹.

If you cannot try it, this is what happens.
A table with the head Number and In full. Rows: 3 × 10⁴, 30 000; 2 × 10⁵, 200 000; 6 × 10⁹, 6 000 000 000.NumberIn full3 × 10⁴30 0002 × 10⁵200 0006 × 10⁹6 000 000 000

A wrong idea many people have

3 × 10⁵ + 4 × 10⁴ = 7 × 10⁹, because you add the front numbers and the powers.

Why it feels right: Adding the powers worked when we multiplied, so it feels right here too.

The test that breaks it: 3 × 10⁵ is 300 000 and 4 × 10⁴ is 40 000. Their sum is 340 000. But 7 × 10⁹ is 7 000 000 000, which is far too big.

What is true: Match the powers first, then add only the front numbers. 3 × 10⁵ + 0.4 × 10⁵ = 3.4 × 10⁵.

A wrong idea many people have

(3 × 10⁴) × (2 × 10⁵) = 6 × 10²⁰, because you multiply the powers.

Why it feels right: The sign between the numbers is times, so it seems everything is multiplied.

The test that breaks it: Work out the sizes: 30 000 × 200 000 = 6 000 000 000, which is 6 × 10⁹. The power 20 would give a 1 followed by 20 zeros.

What is true: Multiply the front numbers. Add the powers, because 10⁴ × 10⁵ = 10⁹.

Key words

power of ten
the 10ⁿ part of a number in standard form, such as 10⁵.

Back to your guess

The answer is B: 6 × 10⁹, by multiplying 3 × 2 and adding the powers 4 + 5.

Multiply the front numbers: 3 × 2 = 6. Add the powers: 4 + 5 = 9, because 10⁴ × 10⁵ = 10⁹. So the answer is 6 × 10⁹. The front numbers are multiplied and the powers are added.

Check yourself

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

  1. Work out (2 × 10³) × (4 × 10⁵).

    Tap your answer.

    1. Not this one. You multiplied the powers. When you multiply numbers with the same base, 10, you add the powers: 3 + 5 = 8. Go to M4.1.1. Try another.

    2. Not this one. You added the front numbers. The two numbers are multiplied, so multiply the front numbers: 2 × 4 = 8. Go to M4.1.5. Try another.

    3. Not this one. You subtracted the powers. That is for division. For a product, add them. Go to M4.1.5. Try another.

    4. Yes, that is right. 2 × 4 = 8 and 3 + 5 = 8. So the answer is 8 × 10⁸.

    See the answer

    8 × 10⁸

    2 × 4 = 8 and 3 + 5 = 8. So the answer is 8 × 10⁸.

    • If you chose A: You multiplied the powers. When you multiply numbers with the same base, 10, you add the powers: 3 + 5 = 8. Go to M4.1.1.
    • If you chose B: You added the front numbers. The two numbers are multiplied, so multiply the front numbers: 2 × 4 = 8. Go to M4.1.5.
    • If you chose C: You subtracted the powers. That is for division. For a product, add them. Go to M4.1.5.

  2. Work out (3 × 10⁶) ÷ (6 × 10²).

    Tap your answer.

    1. Not this one. This is equal to the answer, but 0.5 is below 1, so it is not standard form. Write 0.5 as 5 × 10⁻¹ and lower the power by 1. Go to M4.1.5. Try another.

    2. Yes, that is right. 3 ÷ 6 = 0.5 and 6 − 2 = 4, so the quotient is 0.5 × 10⁴. Then 0.5 = 5 × 10⁻¹, so the answer is 5 × 10³.

    3. Not this one. You divided 6 by 3. The question divides 3 by 6, which is 0.5. Go to M4.1.5. Try another.

    4. Not this one. You moved the point of 0.5 to make 5 but kept the power 4. When the front number is multiplied by 10, the power must go down by 1. Go to M4.1.5. Try another.

    See the answer

    5 × 10³

    3 ÷ 6 = 0.5 and 6 − 2 = 4, so the quotient is 0.5 × 10⁴. Then 0.5 = 5 × 10⁻¹, so the answer is 5 × 10³.

    • If you chose A: This is equal to the answer, but 0.5 is below 1, so it is not standard form. Write 0.5 as 5 × 10⁻¹ and lower the power by 1. Go to M4.1.5.
    • If you chose C: You divided 6 by 3. The question divides 3 by 6, which is 0.5. Go to M4.1.5.
    • If you chose D: You moved the point of 0.5 to make 5 but kept the power 4. When the front number is multiplied by 10, the power must go down by 1. Go to M4.1.5.

  3. Work out (5 × 10⁴) × (6 × 10³).

    Tap your answer.

    1. Not this one. This is equal to the answer, but 30 is not below 10, so it is not standard form. Write 30 as 3 × 10¹. Go to M4.1.5. Try another.

    2. Not this one. You made 30 into 3 but did not raise the power. Dividing the front number by 10 means the power goes up by 1. Go to M4.1.5. Try another.

    3. Yes, that is right. 5 × 6 = 30 and 4 + 3 = 7, giving 30 × 10⁷. Write 30 as 3 × 10¹. Then the power is 7 + 1 = 8: 3 × 10⁸.

    4. Not this one. You added the front numbers. They are multiplied: 5 × 6 = 30. Go to M4.1.5. Try another.

    See the answer

    3 × 10⁸

    5 × 6 = 30 and 4 + 3 = 7, giving 30 × 10⁷. Write 30 as 3 × 10¹. Then the power is 7 + 1 = 8: 3 × 10⁸.

    • If you chose A: This is equal to the answer, but 30 is not below 10, so it is not standard form. Write 30 as 3 × 10¹. Go to M4.1.5.
    • If you chose B: You made 30 into 3 but did not raise the power. Dividing the front number by 10 means the power goes up by 1. Go to M4.1.5.
    • If you chose D: You added the front numbers. They are multiplied: 5 × 6 = 30. Go to M4.1.5.

  4. Work out (4 × 10⁶) + (5 × 10⁵).

    Tap your answer.

    1. Yes, that is right. 5 × 10⁵ = 0.5 × 10⁶. Then 4 + 0.5 = 4.5, so the sum is 4.5 × 10⁶. In full, 4 000 000 + 500 000 = 4 500 000.

    2. Not this one. You added the front numbers without matching the powers. 5 × 10⁵ is only 0.5 × 10⁶, not 5 × 10⁶. Go to M4.1.5. Try another.

    3. Not this one. You added the powers. That is for multiplying. For adding, match the powers and then add only the front numbers. Go to M4.1.5. Try another.

    See the answer

    4.5 × 10⁶

    5 × 10⁵ = 0.5 × 10⁶. Then 4 + 0.5 = 4.5, so the sum is 4.5 × 10⁶. In full, 4 000 000 + 500 000 = 4 500 000.

    • If you chose B: You added the front numbers without matching the powers. 5 × 10⁵ is only 0.5 × 10⁶, not 5 × 10⁶. Go to M4.1.5.
    • If you chose C: You added the powers. That is for multiplying. For adding, match the powers and then add only the front numbers. Go to M4.1.5.

  5. Mount Everest is about 8.8 × 10³ m high and a sand grain is about 5 × 10⁻⁴ m wide. About how many times bigger is the mountain?

    A vertical scale headed Sizes in metres. The ticks, from the bottom to the top, are 10⁻⁶, 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, 1, 10¹, 10², 10³ and 10⁴, each ten times bigger than the one below. Black dots mark sand grain and Mount Everest, each at its true height on the scale.Sizes in metres10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹110¹10²10³10⁴sand grainMount Everest
    Tap your answer.

    1. Not this one. You added 3 and −4. A comparison of size is a division, so subtract the powers: 3 − (−4) = 7. Go to M4.1.5. Try another.

    2. Not this one. You subtracted the powers the wrong way round: −4 − 3 = −7. Take the sand grain's power from the mountain's: 3 − (−4) = 7. The mountain is bigger, so the power must be positive. Go to M4.1.5. Try another.

    3. Yes, that is right. Divide the sizes: 8.8 ÷ 5 = 1.76 and 3 − (−4) = 7. So the mountain is about 1.76 × 10⁷ times bigger.

    See the answer

    About 1.76 × 10⁷ times bigger

    Divide the sizes: 8.8 ÷ 5 = 1.76 and 3 − (−4) = 7. So the mountain is about 1.76 × 10⁷ times bigger.

    • If you chose A: You added 3 and −4. A comparison of size is a division, so subtract the powers: 3 − (−4) = 7. Go to M4.1.5.
    • If you chose B: You subtracted the powers the wrong way round: −4 − 3 = −7. Take the sand grain's power from the mountain's: 3 − (−4) = 7. The mountain is bigger, so the power must be positive. Go to M4.1.5.

  6. From an earlier chapter

    Write 4 500 000 in standard form. What is the power of ten?

    See the answer

    6

    The point moves 6 places left, to after the 4. So 4 500 000 = 4.5 × 10⁶.

  7. Work out (3 × 10⁸) ÷ (6 × 10³) in standard form. What is the power of ten?

    See the answer

    4

    3 ÷ 6 = 0.5 and 8 − 3 = 5. The front number is below 1, so write 0.5 as 5 × 10⁻¹. The answer is 5 × 10⁴.

  8. Work out (2 × 10⁵) + (3 × 10⁴) in standard form. What is the first number?

    See the answer

    2.3

    3 × 10⁴ = 0.3 × 10⁵. Then 2 + 0.3 = 2.3, so the sum is 2.3 × 10⁵.

  9. A row of cells is 5 × 10⁻³ m long. Each cell is 1 × 10⁻⁵ m wide. How many cells are in the row?

    See the answer

    500

    Divide the lengths: 5 ÷ 1 = 5 and −3 − (−5) = 2. So the number of cells is 5 × 10² = 500.

Keep it

Copy this into your exercise book.

Multiply: multiply the front numbers, add the powers. Divide: divide, subtract. Add: match the powers first. Then check the front number is from 1 to below 10.

Draw: (3 × 10⁴) × (2 × 10⁵) in two boxes, 3 × 2 = 6 and 10⁴ × 10⁵ = 10⁹, joined as 6 × 10⁹.

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

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