 # Order Indices, Roots and Reciprocals

In this worksheet, students will order a variety of numbers and expressions according to value, including indices, roots and reciprocals. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Number Operations and Integers

Curriculum subtopic:   Structure and Calculation, Combining Arithmetic Operations

Difficulty level:   ### QUESTION 1 of 10

When we are ordering calculations, we have to be able to do this with all sorts of numbers.

In this activity, we're going to look at doing this with indices, roots and reciprocals.

Definitions

Indices: Indices are numbers written in index form. For example, 22 and 43 are indices.

Roots: Refers to square roots, i.e. a value that, when multiplied by itself, gives the number. For example, √49 is 7.

Reciprocals: These are numbers that are written as 1 over another number. For example, 1/5 and 1/3.

e.g. Put the numbers √81, 1/3 and 23 in ascending order.

A key point here (and in all ordering questions) is that we have been asked for ascending order, which means we are going from smallest to largest.

To order any numbers by size, we need to get them all into the same format so we can compare them.

In this case, the easiest way to do this to calculate to remove the symbols:

23 = 2 x 2 x 2 = 8

√81 = 9

1/3 = 0.3333

From these values, we can now see that the correct order is: 1/3, 23, √81

In this activity, you will order a variety of numbers and expressions according to value, including indices, roots and reciprocals.

Complete the sentence below to describe the golden rule when comparing numbers and expressions.

Which is larger?

 1 5
or
 1 4
 1 5

 1 4

Which is larger?

23 or 32

23

32

Match the indices, roots and reciprocals to their number values.

## Column B

34
81
27
5
√ 25
0.125
1/8
128

Which of the options below has the greatest value?

14

42

√ 144

Put the fractions below into ascending order.

For the smallest, match it to the number 1 and the largest to number 4.

## Column B

1/9
1
1/5
4
1/4
2
1/3
3

Which of the options in the list are equal to 64?

43

26

82

322

If we ordered the three expressions below, which one would be in the middle?

51, √36, 22

51

√ 36

22

Which of the numbers below will come first if we put them in descending order?

24,√81, 19

24

√ 81

19

Match the pairs of indices.

## Column B

34
42
24
32
43
82
91
92
• Question 1

Complete the sentence below to describe the golden rule when comparing numbers and expressions.

EDDIE SAYS
When we are ordering, we need to be able to compare magnitude or value. In order to compare, we must change the values into the same form.
• Question 2

Which is larger?

 1 5
or
 1 4
 1 4
EDDIE SAYS
There's a couple of ways we could approach this question: 1) In a reciprocal, the smaller the denominator then the larger the number. So we can intuitively know that 1/4 is larger. 2) Convert them both to decimals. 1/5 = 0.2 1/4 = 0.25
• Question 3

Which is larger?

23 or 32

32
EDDIE SAYS
To solve this one, we need to evaluate (work out) the values of each expression: 23 = 2 x 2 x 2 = 8 32 = 3 x 3 = 9 We know that 9 is larger than 8, so 32 is greater.
• Question 4

Match the indices, roots and reciprocals to their number values.

## Column B

34
81
27
128
√ 25
5
1/8
0.125
EDDIE SAYS
This one's just a case of remembering those definitions: Index - (or power) is the number of times you multiply the whole number by itself. Root - What number do I times by itself to get this? Reciprocal - It's a fraction, so divide the numerator by the denominator.
• Question 5

Which of the options below has the greatest value?

42
EDDIE SAYS
We need to work out the numbers represented by these expressions and then say which is the biggest. 42 = 4 x 4 = 16 √ 144 = 12 (12 x 12 = 144) Out of 14, 16 and 12, 16 has the greatest value.
• Question 6

Put the fractions below into ascending order.

For the smallest, match it to the number 1 and the largest to number 4.

## Column B

1/9
1
1/5
2
1/4
3
1/3
4
EDDIE SAYS
Remember our rule for reciprocals: "The larger the number on the denominator, the smaller the value." So if we compare the denominators, we know that 1/9 is the smallest and 1/3 is the largest. We can also convert them to decimals: 1 ÷ 9 = 0.111 1 ÷ 5 = 0.2 1 ÷ 4 = 0.25 1 ÷ 3 = 0.333
• Question 7

Which of the options in the list are equal to 64?

43
26
82
EDDIE SAYS
There is usually more than one way to get to a whole number from a power. The large number represents the value which is being multiplied by itself, and the small number represents the amount of times you repeat this. 43 = 4 x 4 x 4 = 64 26 = 2 x 2 x 2 x 2 x 2 x 2 = 64 82 = 8 x 8 = 64 322 = 32 x 32 = 1024 Did you notice the one that was incorrect? Great focus, if you did. Some students forget what 322 actually means, and just do 32 x 2 instead of 32 x 32. This is another common mistake you must be wary of making in an exam!
• Question 8

If we ordered the three expressions below, which one would be in the middle?

51, √36, 22

51
EDDIE SAYS
Once again, this is just a case of working out the whole-number value of each option. The tricky one here is: 51 This means the same as 5 x 1 = 5 √ 36 = 6 22 = 4 The three expressions work out as 4, 5 and 6, so 5 is in the middle.
• Question 9

Which of the numbers below will come first if we put them in descending order?

24,√81, 19

19
EDDIE SAYS
Did you notice the switch here? This question asks for descending order, so we're looking for the highest number to be first. Always take your time to read the question closely to avoid slips of wording like this. 24 = 2 x 2 x 2 x 2 = 16 √ 81 = 9 19 The highest number is 19.
• Question 10

Match the pairs of indices.

## Column B

34
92
24
42
43
82
91
32
EDDIE SAYS
For this question, work out each number value and match those that are the same. 34 = 3 x 3 x 3 x 3 = 81 92 = 9 x 9 = 81 So these two would have to go together. 24 = 2 x 2 x 2 x 2 = 16 42 = 4 x 4 = 16 43 = 4 x 4 x 4 = 64 82 = 8 x 8 = 84 91 = 9 32 = 3 x 3 = 9 Great work, you've completed another activity! Why not learn more about whichever from roots, indices and reciprocals which you found most challenging to work out?
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