Chapter 8 of 8

All 8 chapters
  1. 1Below zero, readunderstood
  2. 2Calculating with negative numbers, readunderstood
  3. 3Multiples and factors, readunderstood
  4. 4Quick tests for dividing, readunderstood
  5. 5Prime numbers, readunderstood
  6. 6Prime factors, common factors and multiples, readunderstood
  7. 7Squares, cubes and roots, readunderstood
  8. 8Powers of ten, readunderstood

M3.1.8 Chapter 8

Powers of ten

How many zeros are there in ten million, and is there a short way to write it?

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

  • Write 1 000 as 10³ and 1 000 000 as 10⁶
  • Say 10 000 000 as ten million or one crore
  • Use powers of ten to compare very big numbers

Sipho reads that a village water tank holds 100 000 litres. A pond holds 10 000 000 litres. Counting the zeros one by one is slow, and you may lose count. A power of ten is a short way to write it.

You know that 10 × 10 = 100 and 10 × 10 × 10 = 1 000. In chapter 7 you wrote 2 × 2 × 2 as 2³. In the same way, 10 × 10 × 10 = 10³ = 1 000, and it has three zeros.

A power of ten is 10 with an index. The index says how many 10s are multiplied. It also tells you how many zeros follow the 1.

Draw it first

Count the zeros in three powers of ten

  1. Boxes with one digit in each box. Step 1: 10³ is a 1 and 3 zeros, with a bracket labelled 3 zeros. Step 2: 10⁶ is a 1 and 6 zeros, with a bracket labelled 6 zeros. Step 3: 10⁷ is a 1 and 7 zeros, with a bracket labelled 7 zeros.10³10003 zeros

    1 1 000 is 10³. It is a 1 and 3 zeros, and the index is 3.

  2. Boxes with one digit in each box. Step 1: 10³ is a 1 and 3 zeros, with a bracket labelled 3 zeros. Step 2: 10⁶ is a 1 and 6 zeros, with a bracket labelled 6 zeros. Step 3: 10⁷ is a 1 and 7 zeros, with a bracket labelled 7 zeros.10⁶10000006 zeros

    2 1 000 000 is 10⁶. It is a 1 and 6 zeros, and the index is 6.

  3. Boxes with one digit in each box. Step 1: 10³ is a 1 and 3 zeros, with a bracket labelled 3 zeros. Step 2: 10⁶ is a 1 and 6 zeros, with a bracket labelled 6 zeros. Step 3: 10⁷ is a 1 and 7 zeros, with a bracket labelled 7 zeros.10⁷100000007 zeros

    3 10 000 000 is 10⁷. It is a 1 and 7 zeros, and the index is 7.

Worked example

Write 1 000 000 as a power of ten.

  1. Count the zeros after the 1. There are 6 of them.
  2. The index is the number of zeros. So the index is 6.
  3. 1 000 000 = 10⁶.

Answer: 10⁶

A table with the head power and number. The rows are 10¹ and 10; 10² and 100; 10³ and 1 000; 10⁴ and 10 000; 10⁵ and 100 000; 10⁶ and 1 000 000; 10⁷ and 10 000 000.powernumber10¹1010²10010³1 00010⁴10 00010⁵100 00010⁶1 000 00010⁷10 000 000
Each step down the table puts one more zero on the number.

Each time the index goes up by 1, the number gains one zero. It is 10 times as big.

Big numbers have names. In some places, 100 000 is called one lakh and 10 000 000 is called one crore. In other places 10 000 000 is ten million. All of these names mean the same numbers.

A table with the head number and name. The rows are 1 000 and thousand; 10 000 and ten thousand; 100 000 and hundred thousand; 1 000 000 and million; 10 000 000 and ten million.numbername1 000thousand10 000ten thousand100 000hundred thousand1 000 000million10 000 000ten million
One lakh is 100 000, which is a hundred thousand. One crore is 10 000 000, which is ten million.

One crore is 100 lakh, because 100 × 100 000 = 10 000 000. In powers, 10⁷ = 10² × 10⁵. The indices add: 2 + 5 = 7.

Worked example

Say 10 000 000 in words.

  1. Count the zeros after the 1. There are 7 of them, so 10 000 000 = 10⁷.
  2. Count all the digits in 10 000 000. There are 8 digits.
  3. Group the digits in threes from the right: 10, 000, 000.
  4. Say it as ten million. Some places say one crore for the same number.

Answer: ten million, or one crore

To compare big numbers, look at the index first. A bigger index means more zeros, so a bigger number. 10⁷ is bigger than 10⁵.

How much bigger? 10⁷ has two more zeros than 10⁵. So 10⁷ is 100 times 10⁵. Two more zeros means 10 × 10 times as big.

A table with the head written and number. The rows are 3 × 10⁶ and 3 000 000; 8 × 10⁵ and 800 000.writtennumber3 × 10⁶3 000 0008 × 10⁵800 000
The same two numbers, written with a power of ten and in full.

Worked example

Which is bigger, 3 × 10⁶ or 8 × 10⁵?

  1. 3 × 10⁶ = 3 × 1 000 000 = 3 000 000.
  2. 8 × 10⁵ = 8 × 100 000 = 800 000.
  3. Count the digits. 3 000 000 has 7 digits. 800 000 has 6 digits.
  4. More digits means a bigger number, so 3 000 000 is bigger.

Answer: The bigger one is 3 × 10⁶, which is 3 000 000.

See it three ways

With things

A truck carries 1 000 sacks. Each sack holds 1 000 seeds. How many seeds are on the truck?

As a picture

Boxes with one digit in each box. 10⁶ is a 1 and 6 zeros, with a bracket labelled 6 zeros.10⁶10000006 zeros

In symbols

10³ × 10³ = 1 000 000 = 10⁶. The indices add: 3 + 3 = 6.

Try it

Count the zeros as you multiply

You need: a sheet of paper, a pencil.

  1. Write 1 at the top of the page.
  2. Under it, write 1 × 10 = 10. Then multiply the answer by 10 again, and again.
  3. Each line, count the zeros in the answer and write the index beside it.
  4. Stop when you reach 100 000. How many zeros has it?

What you will see: Each time you multiply by 10, one more zero arrives. 100 000 has 5 zeros, so it is 10⁵. That is one lakh.

If you cannot try it, this is what happens.
Boxes with one digit in each box. 10⁵ is a 1 and 5 zeros, with a bracket labelled 5 zeros.10⁵1000005 zeros

A wrong idea many people have

10³ means 10 × 3, so 10³ = 30.

Why it feels right: The small 3 sits next to the 10, and multiplying 10 by 3 is quick to do.

The test that breaks it: Write 10 × 10 × 10. First 10 × 10 = 100, then 100 × 10 = 1 000. There are three 10s, and the answer has three zeros. It is not 30.

What is true: The index counts how many 10s you multiply. It does not multiply 10 by that number. 10³ = 1 000.

Key words

power of ten
the number 10 with an index, such as 10³. It is a 1 followed by that many zeros.
lakh
a name used in some places for 100 000.
crore
a name used in some places for 10 000 000, which is ten million.

Back to your guess

The answer is C: 10⁶ is a 1 followed by six zeros, so it is 1 000 000.

10⁶ means six 10s multiplied together. That is a 1 followed by six zeros, or 1 000 000. The small number counts the zeros.

Check yourself

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

  1. What is 10³?

    Tap your answer.

    1. Not this one. You may have multiplied 10 by 3. The index 3 says there are three 10s multiplied together, and three zeros in the answer. Go to M3.1.8. Try another.

    2. Not this one. You may have written the 3 and then two zeros. The index is not a digit of the answer. It counts the zeros after the 1. Go to M3.1.8. Try another.

    3. Yes, that is right. 10³ = 10 × 10 × 10 = 1 000. It has 3 zeros.

    See the answer

    1 000

    10³ = 10 × 10 × 10 = 1 000. It has 3 zeros.

    • If you chose A: You may have multiplied 10 by 3. The index 3 says there are three 10s multiplied together, and three zeros in the answer. Go to M3.1.8.
    • If you chose B: You may have written the 3 and then two zeros. The index is not a digit of the answer. It counts the zeros after the 1. Go to M3.1.8.

  2. Write 100 000 as a power of ten.

    Tap your answer.

    1. Not this one. You counted only four zeros. Look again: 100 000 has five zeros after the 1. Go to M3.1.8. Try another.

    2. Yes, that is right. There are five zeros after the 1, so the index is 5.

    3. Not this one. You counted the digits. 100 000 has six digits, but the index counts the zeros after the 1, and there are five. Go to M3.1.8. Try another.

    See the answer

    10⁵

    There are five zeros after the 1, so the index is 5.

    • If you chose A: You counted only four zeros. Look again: 100 000 has five zeros after the 1. Go to M3.1.8.
    • If you chose C: You counted the digits. 100 000 has six digits, but the index counts the zeros after the 1, and there are five. Go to M3.1.8.

  3. Which of these numbers is one crore?

    Tap your answer.

    1. Yes, that is right. One crore is ten million, 10 000 000. It is 1 followed by seven zeros, or 10⁷.

    2. Not this one. That is one million, which is ten lakh. One crore has seven zeros. Go to M3.1.8. Try another.

    3. Not this one. That is one lakh, which is one hundred thousand. One crore is 100 times as big. Go to M3.1.8. Try another.

    See the answer

    10 000 000

    One crore is ten million, 10 000 000. It is 1 followed by seven zeros, or 10⁷.

    • If you chose B: That is one million, which is ten lakh. One crore has seven zeros. Go to M3.1.8.
    • If you chose C: That is one lakh, which is one hundred thousand. One crore is 100 times as big. Go to M3.1.8.

  4. Which is bigger, 4 × 10⁶ or 9 × 10⁵?

    Tap your answer.

    1. Not this one. You looked at the 9, which is bigger than 4. But 10⁶ is ten times as big as 10⁵. Write both in full: 4 000 000 and 900 000. Go to M3.1.8. Try another.

    2. Not this one. The index is not multiplied by the front number. It counts the zeros. Write both in full: 4 000 000 and 900 000. Go to M3.1.8. Try another.

    3. Yes, that is right. 4 × 10⁶ = 4 000 000 and 9 × 10⁵ = 900 000. 4 000 000 has seven digits and 900 000 has six.

    See the answer

    4 × 10⁶

    4 × 10⁶ = 4 000 000 and 9 × 10⁵ = 900 000. 4 000 000 has seven digits and 900 000 has six.

    • If you chose A: You looked at the 9, which is bigger than 4. But 10⁶ is ten times as big as 10⁵. Write both in full: 4 000 000 and 900 000. Go to M3.1.8.
    • If you chose B: The index is not multiplied by the front number. It counts the zeros. Write both in full: 4 000 000 and 900 000. Go to M3.1.8.

  5. One crore is 10 000 000 and one lakh is 100 000. How many lakh make one crore?

    Tap your answer.

    1. Not this one. 1 000 lakh is 100 000 000, which is ten times too big. Divide 10 000 000 by 100 000. Go to M3.1.8. Try another.

    2. Yes, that is right. 10 000 000 ÷ 100 000 = 100. In powers, 10⁷ ÷ 10⁵ = 10², which is 100.

    3. Not this one. Ten lakh is 1 000 000, which is one million. One crore is ten million, 10 times as big. Go to M3.1.8. Try another.

    See the answer

    100 lakh

    10 000 000 ÷ 100 000 = 100. In powers, 10⁷ ÷ 10⁵ = 10², which is 100.

    • If you chose A: 1 000 lakh is 100 000 000, which is ten times too big. Divide 10 000 000 by 100 000. Go to M3.1.8.
    • If you chose C: Ten lakh is 1 000 000, which is one million. One crore is ten million, 10 times as big. Go to M3.1.8.

  6. Write 10⁴ as an ordinary number.

    See the answer

    10 000

    10⁴ is a 1 followed by 4 zeros. That is 10 × 10 × 10 × 10 = 10 000.

  7. Write 3 × 10⁴ as an ordinary number.

    See the answer

    30 000

    10⁴ = 10 000, so 3 × 10⁴ = 3 × 10 000 = 30 000.

  8. From an earlier chapter

    Find the square root of 121.

    See the answer

    11

    11 × 11 = 121, so the square root of 121 is 11.

Keep it

Copy this into your exercise book.

10³ = 1 000, 10⁶ = 1 000 000. The index counts the zeros. 10 000 000 is ten million, or one crore. 100 000 is one hundred thousand, or one lakh. A bigger index means a bigger number.

Draw: Boxes for 10⁷: a 1 and seven zeros, grouped 10, 000, 000. Draw a bracket under the zeros and write 7 zeros.

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

No internet tomorrow? Save the whole book as one file (805 KB). It opens on any phone.