 # Estimate Powers and Roots

In this worksheet, students will estimate the answers to squared powers and roots so that they are able to work these out without a calculator. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Indices and Surds

Curriculum subtopic:   Structure and Calculation, Powers and Roots

Difficulty level:   ### QUESTION 1 of 10

When we aren't using a calculator, it can be very hard to reach exact correct answers to questions and often this level of detail is not required anyway.

In these instances, we need to estimate answers.

Estimating means making an educated guess.

In this activity, we're going to look at how we do this with square numbers and square roots, which can often give long and complex answers when calculated.

Square numbers:

To be able to estimate with numbers of this type, we must know our square numbers up to 102.

These are: 1,4,9,16, 25, 36, 49, 64, 81, 100.

e.g. Estimating a square root

Between which two integers does √80 lie?

To do this, we need to look at which square numbers are closest to 80.

We can see it is between 64 and 81.

As √64 = 8 and √81 = 9, √80 must be between 8 and 9.

As 80 is closer to 81 than 64, √80 will be very close to 9.

e.g. Estimating a square number

Between which two squares will 7.12 lie?

Again, we need to look at our square numbers.

7.1 is between 7 and 8.

72 = 49, 82 = 64

Therefore, 7.12 must be between 49 and 64.

In this activity, we will estimate the answers to squared powers and roots so that we are able to work these out without a calculator.

This will be very useful for your mental maths, speed in solving problems, and checking that your answers are feasible.

Match each square number below to its square.

## Column B

52
4
62
64
82
25
22
36

Match each square root below with their positive value.

## Column B

√ 25
5
√ 1
1
√ 81
9
√ 100
10

√67 will be between which two integers?

## Column B

√ 25
5
√ 1
1
√ 81
9
√ 100
10

3.72 will be between which two square numbers?

## Column B

√ 25
5
√ 1
1
√ 81
9
√ 100
10

Imagine that your friend is working on this calculation:

8.22

They have created the three lines of working below.

Select if each line is correct or not by ticking a box in each row.

Imagine that your friend is working on this calculation:

√40

They have created the three lines of working below.

Select if each line is correct or not by ticking a box in each row.

Between which two numbers does √99 lie?

7 and 8

8 and 9

9 and 10

Find an estimate for √42.

7 and 8

8 and 9

9 and 10

Is √15 closer to 3 or 4?

3

4

8.2

3

4

• Question 1

Match each square number below to its square.

## Column B

52
25
62
36
82
64
22
4
EDDIE SAYS
Remember that square means a number multiplied by itself. e.g. 32 = 3 x 3 = 9 It is really useful to commit at least the first 10 square numbers and their squares to memory so that you don't need to calculate these each time and can just recognise them.
• Question 2

Match each square root below with their positive value.

## Column B

√ 25
5
√ 1
1
√ 81
9
√ 100
10
EDDIE SAYS
To find a square root, ask yourself the question: "What do I need to times by itself to reach this number?" e.g. 36 = 6 x 6 So, √36 = 6 If you know the most common square numbers, then square roots become much easier to spot too.
• Question 3

√67 will be between which two integers?

EDDIE SAYS
Which two square numbers is 67 between? It is between 64 and 81. 64 is the product of 82 81 is the product of 92 Therefore, the answer to √67 must be between 8 and 9. Can you reason whether it is closer to 8 or 9 for an extra challenge?
• Question 4

3.72 will be between which two square numbers?

EDDIE SAYS
3.7 is between 3 and 4. 32 = 9 42 = 16 So 3.72 must be between 9 and 16. Which do you think it is closest to? This is good practise, let's keep going!
• Question 5

Imagine that your friend is working on this calculation:

8.22

They have created the three lines of working below.

Select if each line is correct or not by ticking a box in each row.

EDDIE SAYS
Your friend's big mistake here is in line 2. 82 is indeed 64 but what is 92? Your friend has done the sum 9 x 2, rather than 9 x 9 - an easy mistake to make! This error then follows through to the last line of their working, as 8.22 is actually between 64 and 81. In fact 8.2 x 8.2 = 67.24 As 8.2 is closer to 8 than 9, it makes sense that this answer is closer to 64 than 81, doesn't it?
• Question 6

Imagine that your friend is working on this calculation:

√40

They have created the three lines of working below.

Select if each line is correct or not by ticking a box in each row.

EDDIE SAYS
If you are estimating a square root, can you ever say exactly what the answer is? No! So your friend has definitely made an error here. Their first two lines of working are correct, as the answer will be between 36 and 49. The final line is the only place where their working has gone wrong.
• Question 7

Between which two numbers does √99 lie?

9 and 10
EDDIE SAYS
Remember to work out the square numbers which 99. 99 is between 81 and 100. √81 = 9 √100 = 10 So √99 is between 9 and 10. Which will it be closest to, do you think?
• Question 8

Find an estimate for √42.

EDDIE SAYS
42 is between 36 and 49. √36 = 6 √49 = 7 We can see that √42 must lie somewhere between 6 and 7. The centre of these two numbers is between 42 and 43, which would correlate to around 6.5 42 is around the middle of 36 and 49, but a bit closer to 36. Therefore, we can assume that the answer will be slightly less that 6.5 So any answer between 6.3 and 6.6 is acceptable. The exact answer is 6.481
• Question 9

Is √15 closer to 3 or 4?

4
EDDIE SAYS
We can see that √15 is between √9 (3) and √16 (4). As a number, 15 is much closer to 16 than 9, so the root must be closer to 4. This should be beginning to make sense now. If you're not feeling confident, go back to the Introduction page and make some notes for your revision.
• Question 10

8.2

EDDIE SAYS
We can see that 8.2 is between 8 and 9. So the answer to 8.22 is between 64 and 81. As 8.2 is closer to 8 than 9, 8.22 is going to be closer to 82 (64) than 92 (81). Therefore, you can give any value from 66-69 as your answer. The exact answer is 67.24 Well done, that’s another activity completed!
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