3. Index notation
We use index notation as shorthand for multiplication by the same number.
x × x × x × x × x
= x5
For example:
index or power x5
base
This number is read as ‘x to the power of 5’.
Teacher notes
Start by asking pupils to simplify x + x + x + x + x. This is 5 lots of x, which is written as 5x.
Next ask pupils how we could simplify x × x × x × x × x. Make sure there is no confusion between this repeated multiplication and the previous example of repeated addition.
If x is equal to 2, for example, x + x + x + x + x equals 10, while x × x × x × x × x equals 32.
Some pupils may suggest writing xxxxx. While this is not incorrect, neither has it moved us on very far. Point out the problems of readability, especially with high powers.
When we write a number or term to the power of another number it is called index notation. The power, or index (plural indices), is the superscript number, in this case 5. The number or letter that we are multiplying successive times, in this case, x, is called the base.
Practice the relevant vocabulary: x2 is read as ‘x squared’ or ‘x to the power of 2’; x3 is read as ‘x cubed’ or ‘x to the power of 3’; x4 is read as ‘x to the power of 4’.
y × y × y
= y3
“y to the power of 3” or “y cubed”
z × z
= z2
“z to the power of 2” or “z squared”
q × q × q × q
= q4
“q to the power of 4”

4. Using index notation
Using your knowledge of index notation, can you simplify the following expressions?
3p × 2p =
3 × p × 2 × p =
6p2
q2 × q3 =
q × q × q × q × q = q5
3r × r2 =
3 × r × r × r =
3r3
Teacher notes
Discuss each example briefly and remind pupils to multiply any numbers together first followed by letters in alphabetical order.
In the last example, 3t × 3t, the use of brackets may need further clarification. We must put a bracket around the 3t since both the 3 and the t are squared. If we wrote 3t 2, then only the t would be squared.
For example, if t was 2 then 3t would be equal to 6. We would then have 62, 36. If we wrote 3t 2, that would mean 3 × 22 or 3 × 4 which is 12.
Remember the order of operations – BIDMAS. Brackets are worked out before indices, but indices are worked out before multiplication.
3t × 3t =
(3t)2
= 3 × 3 × t × t
= 9t2

6. Multiplying terms with the same base
When we multiply two terms with the same base, the indices are added together.
For example,
a4 × a2 =
(a × a × a × a) × (a × a)
= a × a × a × a × a × a
= a6
4 + 2 = 6
Teacher notes
Stress that the indices can only be added when the base is the same.
In general, the rule when multiplying two terms with the same base is:
xm × xn = x(m + n)

7. Multiplying indices

8. Marking homework
Simon has got his homework wrong! How would you explain his mistakes and what are the correct answers?
a4 x a2 = a8
b3 x b2 = b6
3a2 x 3a5 = 6a7
4b2 x 2b3 = 6b6
5y2 x 4y4 = 9y8
(a × a × a × a) × (a × a) = a6
(b × b × b) × (b × b) = b5
3 × (a × a) × 3 × (a × a × a × a × a)
= 9a7
Teacher notes
For each question, encourage the students to see if they can spot the mistakes that have been made by Simon and state how they would rectify these mistakes.
The indices have been multiplied together. They should have been added.
The bases have been added together. They should have been multiplied.
The bases have been added when they should have been multiplied and the indices have been multiplied when they should have been added.
Again, the bases have been added when they should have been multiplied and the indices have been multiplied when they should have been added.
4 × (b × b) × 2 × (b × b × b) =
8b5
5 × (y × y) × 4 × (y × y × y × y)
= 20y6

9. Dividing terms with the same base
When we divide two terms with the same base, the indices are subtracted.
For example,
5 – 2 = 3
a × a × a × a × a
a × a
a5 ÷ a2 = =
a × a × a = a3
6 – 4 = 2 2
4 × p × p × p × p × p × p
2 × p × p × p × p
4p6 ÷ 2p4 = =
2 × p × p =
2p2
Teacher notes
Stress that the indices can only be subtracted when the base is the same.
In general, the rule when dividing two terms with the same base is:
xm ÷ xn = x(m – n)

10. Dividing indices

12. Expressions of the form (xm)n
When a term is in the form (xm)n the m and n can be multiplied.
For example,
(y3)2 =
y3 × y3
(q2)4 =
q2 × q2 × q2 × q2
= (y × y × y) × (y × y × y)
= q ( )
= y6
= q8
In general, the rule when dealing with a single term contained within a bracket is:
(xm)n = xmn

13. Expressions of the form (xy)n
When a term is in the form (xy)n the n can be applied to both the x and y.
For example,
(xy)3 =
xy × xy × xy
(pq2)4 =
pq2 × pq2 × pq2 × pq2
= (x × x × x) × (y × y × y)
= p( )q ( )
= x3y3
= p4q8
In general, the rule when dealing with two or more terms contained within brackets is:
(xy)n = xnyn 13

14. Brackets and indices

15. The zero index
It is possible to write a term to the power of zero. This is known as the zero index.
Can you write an equivalent expression for the term x0?
Using the rule that: xm ÷ xn = x(m – n)
x4 ÷ x4 =
x(4 – 4) = x0
If x is non-zero then:
x4 ÷ x4 = 1
And so,
Teacher notes
Stress that this rule is only true for non-zero values of x. 00 is undefined.
x0 = 1
In general, the rule for a non-zero value of x raised to the power of zero is:
x0 = 1 (if x ≠ 0)

17. Summary of the index laws

18. Bacterial growth
Grotty green and rotten red are types of bacteria. Both of them multiply by dividing in two, but they do this at different rates.
The Grotty greens divide every 4 minutes.
The Rotten reds divide every 5 minutes.
After one hour, how many bacteria will there be of each type?
Teacher notes
In one hour the Grotty greens will double (60mins ÷ 4mins) 15 times. Therefore, there will be 215 = Grotties after one hour.
The Rotten reds will double (60mins ÷ 5mins) = 12 times. Therefore there will be 212 = 4096 Rottens after an hour.
After 2 hours there will be 230 Grotties and 224 Rottens. This means there will be 230 ÷ 224 = = 26 = 64 times more greens than reds.
After 2 hours, how many times bigger is the population of the greens compared to the reds?

19. Grains of rice and the chess board
An ancient Indian mathematician invented chess and showed his creation to the ruler of the country.
The ruler was so pleased, he gave the inventor the right to name his prize.
The mathematician asked the ruler for this: One grain of rice on the 1st square of the board, 2 on the 2nd square, 4 on the 3rd square and so on.
The ruler was offended that he asked for so little.
Teacher notes
This question can be as involved or as short as wished. A satisfactory answer might simply be to say that the number of rice grains is greater than 263, which is vast. However, there is also an opportunity to introduce summations of power series to find an exact value for the number of rice grains.
The following description uses some very high level maths and you may find it is inappropriate for the level of your class. It does however give a fascinating insight into the true cost. Finding an approximation to the monetary value of the rice gives practice at the use of index laws:
Define S(n) = …. 2n–1.
Then 2S(n) = … 2n
And so S(n) = 2S(n) – S(n) = 2n – 20 = 2n – 1
There would be S(64) grains of rice on the chessboard, and from above S(64) is approximately 264. There are approximately 30,000 grains of rice in a kilogram, and 30,000 is approximately 215. A kilogram of rice costs approximately 50 p = £2–1. So, the monetary value of the rice on the chessboard will be:
Price of a kilogram of rice*(Number of grains of rice ÷ Number of grains of rice per kilogram) = 2–1 × (264 ÷ 215) = 2(64–15–1) = £248.
This is approximately £300,000 billion worth of rice!
Should he have been? Why/why not?
After the ruler realised he had been tricked, he asked the mathematician to count the rice to ensure it was correct!