Year 7 · Unit 1 Number systems and place value · Lesson 2 of 7

What does the small raised number mean?

A power is repeated multiplication, and a root undoes it. Build 3 to the power 4 one factor at a time, then find the side of a square from its area.

Number and AlgebraNo calculator

Warm-up

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

  1. What is 7 × 7?
    Answer49. It is also the number of tiles in a square of 7 rows of 7.
  2. What is 3 × 3 × 3?
    Answer27. Three threes multiplied together.
  3. What is the value of the 4 in 6.04?
    Answer0.04, four hundredths. The digit is in the hundredths place.
Words used in this lesson
power
a number written with a small raised number, such as 3 with 4 raised. For example: 3 to the power 4 is 81.
base
the number that is multiplied by itself. For example: In 5² the base is 5.
exponent (index)
the raised number that counts the factors. For example: In 5² the exponent is 2.
square number
a whole number multiplied by itself. For example: 36 = 6 × 6.
cube number
a whole number used as a factor three times. For example: 27 = 3 × 3 × 3.
square root
the number that multiplies by itself to give a number. For example: the square root of 49 is 7.
cube root
the number used as a factor three times to give a number. For example: the cube root of 64 is 4.

A power counts the factors

A mosaic floor of 7 rows of 7 tiles has 7 × 7 = 49 tiles. Mathematicians write that as 7², said “7 squared”. In general, a power such as 3⁴ has a base (3) and an exponent (4), and the exponent says how many times the base is used as a factor.

Predict first: what is 3⁴? Many people say 12. Build it one factor at a time and see.

3
4
3⁴the power
81its value

3⁴ means four 3s multiplied: 3 × 3 × 3 × 3 = 81. It does not mean 3 × 4.

Show me the steps
  1. Base 3, exponent 1: one factor, 3¹ = 3.
  2. Exponent 2: 3 × 3 = 9. This is 3 squared, 3² = 9.
  3. Exponent 3: 9 × 3 = 27. This is 3 cubed, 3³ = 27.
  4. Exponent 4: 27 × 3 = 81, so 3⁴ = 81. Not 3 × 4 = 12.
  5. Base 2, exponent 5: 2, 4, 8, 16, 32. 2⁵ = 32.
  6. Base 5, exponent 5: 5, 25, 125, 625, 3125. The values grow very quickly.

A power is repeated multiplication. 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, five factors of 2. Squared means exponent 2 and cubed means exponent 3, because those powers give the area of a square and the volume of a cube.

A root undoes a power. √49 = 7 because 7² = 49, and ∛64 = 4 because 4³ = 64. Numbers such as 49 and 64 are perfect squares (and 64 is also a perfect cube).

Roots as a side or an edge

A square has area 50. How long is its side? The square root of 50 is the side. It is not a whole number: 7 × 7 is 49 and 8 × 8 is 64, so it lies between 7 and 8. Change the area and switch to cube roots.

50
7 and 8the root lies between
49 and 64the squares either side

The square root of 50 is between 7 and 8, because 7² = 49 and 8² = 64.

Show me the steps
  1. Area 49: the 7 by 7 square fits exactly, so the square root of 49 is 7. A perfect square.
  2. Area 50: the 7 by 7 square fits with 1 cell left over. The side is between 7 and 8, and nearer 7.
  3. Area 63: still a 7 by 7 square with 14 cells over, and 63 is close to 64, so the side is nearly 8.
  4. Area 64: an 8 by 8 square fits exactly. The square root of 64 is 8.
  5. Cube root, volume 64: a 4 by 4 by 4 cube fits exactly, so the cube root of 64 is 4.
  6. Cube root, volume 50: the 3 by 3 by 3 cube (27) fits, and 4 by 4 by 4 (64) is too big, so the edge is between 3 and 4.

A common mistake: “2⁴ is 2 × 4 = 8.” The exponent says how many factors, not what to multiply by. 2⁴ = 2 × 2 × 2 × 2 = 16.

Worked example: where does a root lie?

Between which two whole numbers is √70?

  1. List squares near 70: 8² = 64 and 9² = 81.
  2. 70 is between 64 and 81, so √70 is between 8 and 9.
  3. 70 is 6 above 64, out of the 17 between 64 and 81, so try 8.4: 8.4 × 8.4 = 70.56, which is just over 70. So √70 is a little under 8.4 and lies between 8 and 9.

Your go. The first step is given. Between which two whole numbers is ∛40?

  1. List cubes near 40: 3³ = 27. Find the next cube yourself.
  2. Say where 40 sits between them, then check.
Check the answer

4³ = 64, and 27 < 40 < 64, so ∛40 is between 3 and 4.

Check before you go on

1. What is the value of 2⁴?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. What is the value of 4³?

3. What is √144?

Level 2 · Building

4. What is ∛64?

5. A square patio has sides of 13 m. What is its area in square metres?

6. Between which two whole numbers does √50 lie?

Level 3 · Stretch

7. A sheet of paper is folded in half again and again, and every fold doubles the number of layers. What is the smallest number of folds that gives more than 1000 layers?

8. How many perfect squares are there from 1 to 200, counting 1?

Something to think about

Is 1 a perfect square? A perfect cube? Which numbers up to 1000 are both a perfect square and a perfect cube? What do they have in common?

One way to think about it

1 = 1² = 1³, so yes to both. Up to 1000 the numbers that are both are 1, 64 and 729: 64 = 8² = 4³ and 729 = 27² = 9³. They are 1⁶, 2⁶ and 3⁶: a number that is both a square and a cube is a sixth power.

Exit ticket

9. Which statement about 5² and 2⁵ is true?