Chapter 7 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

M3.1.7 Chapter 7

Squares, cubes and roots

What does the small raised 2 in 5² mean?

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 2 × 2 × 2 as 2³, and 5 squared as 5²
  • Find the square root of a square number up to 144
  • Find the cube root of 8, 27 and 64

Ana lays floor tiles in a square, with 5 tiles along each side. How many tiles does she need?

Draw it first

Build a square of 5 tiles by 5 tiles

  1. Square tiles. Step 1: one row of 5 tiles, with 5 written above and 1 on the left. Step 2: 5 rows of 5 tiles, with 5 above and 5 on the left. Step 3: the line 5 × 5 = 25 under the square.51

    1 Lay one row of 5 tiles.

  2. Square tiles. Step 1: one row of 5 tiles, with 5 written above and 1 on the left. Step 2: 5 rows of 5 tiles, with 5 above and 5 on the left. Step 3: the line 5 × 5 = 25 under the square.55

    2 Lay 5 rows of 5 tiles. The tiles make a square.

  3. Square tiles. Step 1: one row of 5 tiles, with 5 written above and 1 on the left. Step 2: 5 rows of 5 tiles, with 5 above and 5 on the left. Step 3: the line 5 × 5 = 25 under the square.555 × 5 = 25

    3 Count the tiles: 5 × 5 = 25. We write 5 × 5 as 5², and say "5 squared".

Multiplying a number by itself gives a square number. 25 is a square number, because it is 5 × 5. We call a number written like 5² a power.

In 5², the small raised 2 is the index. It tells you how many 5s to multiply together. Some books call it the exponent.

Now think of a block made of small cubes. A cube number comes from multiplying three equal whole numbers. For example, 2 × 2 × 2 = 8. We write it 2³ and say "2 cubed".

On the left, one layer of 4 small cubes seen from above, as a square of 2 by 2, labelled one layer: 4. On the right, a cube made of 2 layers of 4 small cubes, labelled two layers: 8.one layer: 4two layers: 8
One layer has 2 × 2 = 4 small cubes. Two layers have 4 + 4 = 8, which is 2 × 2 × 2.

Worked example

Write 3 × 3 × 3 × 3 as a power, and work out its value.

  1. Count the 3s. There are four of them. So the index is 4.
  2. Write 3⁴.
  3. 3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81.

Answer: 3⁴ = 81

Here are the square numbers up to 144. Learn them. They will turn up again and again.

A table of square numbers with the head n, n², n and n². The rows are 1 and 1, and 7 and 49; 2 and 4, and 8 and 64; 3 and 9, and 9 and 81; 4 and 16, and 10 and 100; 5 and 25, and 11 and 121; 6 and 36, and 12 and 144.nn²nn²117492486439981416101005251112163612144
The squares of 1 to 12.

And here are the first cube numbers.

A table of cube numbers with the head n and n³. The rows are 1 and 1; 2 and 8; 3 and 27; 4 and 64; 5 and 125.nn³11283274645125
The cubes of 1 to 5.

Other powers work the same way. 2⁴ = 2 × 2 × 2 × 2 = 16. Squares and cubes are the powers with index 2 and index 3.

Now go backwards. 5² = 25, so the square root of 25 is 5. We write √25 = 5. The square root is the positive number that multiplies by itself to give the number. Also, −5 × −5 = 25, but √25 always means 5.

In the same way, 2³ = 8, so the cube root of 8 is 2. The cube root of a number is the number that gives it when it is cubed.

Worked example

Find the square root of 144.

  1. Look for a number that multiplies by itself to make 144.
  2. 10 × 10 = 100 is too small. 11 × 11 = 121 is still too small.
  3. 12 × 12 = 144. So √144 = 12.

Answer: √144 = 12

Worked example

Find the cube roots of 8, 27 and 64.

  1. 2 × 2 × 2 = 8, so the cube root of 8 is 2.
  2. 3 × 3 × 3 = 27, so the cube root of 27 is 3.
  3. 4 × 4 × 4 = 64, so the cube root of 64 is 4.
  4. So the cube roots of 8, 27 and 64 are 2, 3 and 4.

Answer: 2, 3 and 4

See it three ways

With things

A square floor has 36 tiles in equal rows. How many tiles are along one side?

As a picture

A square of 36 tiles, 6 rows of 6, with 6 written above and 6 on the left.66

In symbols

6 × 6 = 36, so 6² = 36. The square root of 36 is 6, and we write √36 = 6.

Try it

Make a square from stones

You need: 20 small stones, beans or buttons.

  1. Count out 16 stones. Try to lay them in a square, with the same number in each row and column.
  2. Put them back and count out 12 stones. Try again.
  3. Try 9 stones, then 20 stones. Which numbers make a square?

What you will see: The 16 stones make a square of 4 rows of 4. The 9 stones make 3 rows of 3. The 12 and 20 stones do not. Only square numbers make a square.

If you cannot try it, this is what happens.
A square of 16 tiles, 4 rows of 4, with 4 written above and 4 on the left.44

A wrong idea many people have

5² means 5 × 2, so 5² = 10.

Why it feels right: The small 2 looks like a number to multiply by, and 5 and 2 are both there.

The test that breaks it: Make a square of tiles, 5 along each side. Count them. There are 25 tiles, not 10. The same goes for 2³: a 2 by 2 by 2 block has 8 cubes, not 6.

What is true: The index is how many copies of the number you multiply together. 5² = 5 × 5 = 25 and 2³ = 2 × 2 × 2 = 8.

Key words

square number
a number made by multiplying a whole number by itself, such as 25.
cube number
a number made by multiplying three equal whole numbers, such as 2 × 2 × 2 = 8.
power
a number written with an index, such as 5² or 2³.
index
the small raised number. It says how many of the number to multiply together.
square root
the positive number that multiplies by itself to give a number. √25 = 5.
cube root
the number that gives a cube number when it is cubed. The cube root of 8 is 2.

Back to your guess

The answer is B: 5² means 5 × 5, which is 25.

5² means 5 × 5, which is 25. The small number tells you how many 5s to multiply together. It does not multiply the 5 by 2.

Check yourself

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

  1. Write 2 × 2 × 2 × 2 × 2 as a power, and work out its value.

    Tap your answer.

    1. Not this one. You swapped the 2 and the 5. The number being multiplied is 2, and it is used five times, so the index is 5. Go to M3.1.7. Try another.

    2. Not this one. The small number does not multiply the 2. It counts how many 2s are multiplied together. Go to M3.1.7. Try another.

    3. Yes, that is right. There are five 2s, so the index is 5. 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16 and 16 × 2 = 32.

    See the answer

    2⁵ = 32

    There are five 2s, so the index is 5. 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16 and 16 × 2 = 32.

    • If you chose A: You swapped the 2 and the 5. The number being multiplied is 2, and it is used five times, so the index is 5. Go to M3.1.7.
    • If you chose B: The small number does not multiply the 2. It counts how many 2s are multiplied together. Go to M3.1.7.

  2. What is 7²?

    Tap your answer.

    1. Not this one. You added 7 and 2. A power is a multiplying, not an adding. 7² = 7 × 7. Go to M3.1.7. Try another.

    2. Yes, that is right. 7² means 7 × 7, and 7 × 7 = 49.

    3. Not this one. You worked out 7 × 2. The index 2 means two 7s multiplied together: 7 × 7. Go to M3.1.7. Try another.

    See the answer

    49

    7² means 7 × 7, and 7 × 7 = 49.

    • If you chose A: You added 7 and 2. A power is a multiplying, not an adding. 7² = 7 × 7. Go to M3.1.7.
    • If you chose C: You worked out 7 × 2. The index 2 means two 7s multiplied together: 7 × 7. Go to M3.1.7.

  3. What is 4³?

    Tap your answer.

    1. Yes, that is right. 4 × 4 = 16, then 16 × 4 = 64. So 4³ = 64.

    2. Not this one. You may have worked out 4 × 3. The index 3 means three 4s multiplied together: 4 × 4 × 4. Go to M3.1.7. Try another.

    3. Not this one. You stopped after two 4s. That is 4². For 4³ you multiply by 4 once more. Go to M3.1.7. Try another.

    See the answer

    64

    4 × 4 = 16, then 16 × 4 = 64. So 4³ = 64.

    • If you chose B: You may have worked out 4 × 3. The index 3 means three 4s multiplied together: 4 × 4 × 4. Go to M3.1.7.
    • If you chose C: You stopped after two 4s. That is 4². For 4³ you multiply by 4 once more. Go to M3.1.7.

  4. The picture shows 81 tiles in a square. What is the square root of 81?

    A square made of 81 tiles in equal rows and columns.
    Tap your answer.

    1. Not this one. You divided 81 by 3. The square root is the number of tiles along one side, which multiplies by itself to give 81. Go to M3.1.7. Try another.

    2. Not this one. 3 × 3 × 3 × 3 = 81, so 3 is the number in 3⁴ = 81. The square root uses the number twice: 9 × 9 = 81. Go to M3.1.7. Try another.

    3. Yes, that is right. 9 × 9 = 81, so the square root of 81 is 9. It is the number of tiles along one side.

    See the answer

    9

    9 × 9 = 81, so the square root of 81 is 9. It is the number of tiles along one side.

    • If you chose A: You divided 81 by 3. The square root is the number of tiles along one side, which multiplies by itself to give 81. Go to M3.1.7.
    • If you chose B: 3 × 3 × 3 × 3 = 81, so 3 is the number in 3⁴ = 81. The square root uses the number twice: 9 × 9 = 81. Go to M3.1.7.

  5. The picture shows 64 small cubes in one big cube. What is the cube root of 64?

    A cube made of 64 small cubes, with the same number of small cubes along each edge.
    Tap your answer.

    1. Not this one. You halved 64. The cube root is the number of small cubes along one edge, used three times in a multiplying. Go to M3.1.7. Try another.

    2. Yes, that is right. 4 × 4 × 4 = 64, so the cube root of 64 is 4. It is the number of small cubes along one edge.

    3. Not this one. 8 × 8 = 64, so 8 is the square root of 64. The cube root uses the number three times: 4 × 4 × 4 = 64. Go to M3.1.7. Try another.

    See the answer

    4

    4 × 4 × 4 = 64, so the cube root of 64 is 4. It is the number of small cubes along one edge.

    • If you chose A: You halved 64. The cube root is the number of small cubes along one edge, used three times in a multiplying. Go to M3.1.7.
    • If you chose C: 8 × 8 = 64, so 8 is the square root of 64. The cube root uses the number three times: 4 × 4 × 4 = 64. Go to M3.1.7.

  6. Work out 9².

    See the answer

    81

    9² means 9 × 9, and 9 × 9 = 81.

  7. Find the square root of 100.

    See the answer

    10

    10 × 10 = 100, so the square root of 100 is 10.

  8. From an earlier chapter

    Write 72 as a product of primes.

    See the answer

    72 = 2 × 2 × 2 × 3 × 3

    72 = 8 × 9, and 8 = 2 × 2 × 2 and 9 = 3 × 3. With powers this is 2³ × 3².

Keep it

Copy this into your exercise book.

5² = 5 × 5 = 25, a square number. 2³ = 2 × 2 × 2 = 8, a cube number. The square root of 25 is 5. The cube root of 8 is 2. The index counts how many of the number you multiply together.

Draw: A square of 5 by 5 small squares with 25 written under it. Next to it, draw a cube made of 8 small cubes and write 8.

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

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