#### Solved Examples and Worksheet for Approximating Square Roots of Irrational Numbers

Q1Estimate the value of 3619 to the nearest tenth.

A. 6.1
B. 6.3
C. 6.4
D. 6.2

Step: 1
3619
[Given.]
Step: 2
= 193
[Find square root of each number.]
Step: 3
= 6.3333...
[Divide.]
Step: 4
= 6.3
[Round to the nearest tenth.]
Q2What is the integer which is nearest to 38?

A. 5
B. 6
C. 8
D. 7

Step: 1
An integer should not have a decimal part. So, the integer nearest to 38 will be the square root of the perfect square nearest to 38.
Step: 2
The perfect square that is nearest to 38 is 36.
Step: 3
So, the integer which is nearest to 38 is 36 = 6.
Q3What is the integer nearest to 10?

A. 5
B. 4
C. 3
D. 2

Step: 1
10 is not a perfect square as 10 does not result in an integer value.
Step: 2
The integer nearest to 10 is the square root of the perfect square nearest to 10.
[An integer must not have a decimal part.]
Step: 3
The perfect square, which is nearest to 10 is 9.
[9 = 3, which is an integer.]
Step: 4
So, the integer nearest to 10 is 9 = 3
Q4What is the approximate value of 22?

A. 8
B. 6
C. 5
D. 9
E. 7

Step: 1
The perfect squares nearest to 22 are 16 and 25.
Step: 2
So, the value of 22 must lie between 4 and 5.
Step: 3
The number 22 is closer to 25 than 16.
Step: 4
So, the approximate value of 22 = 5.
Q5What is the approximate value of 22?

A. 10
B. 6
C. 7
D. 5
E. 8

Step: 1
The perfect squares nearest to 22 are 16 and 25.
Step: 2
So, the value of 22 must lie between 4 and 5.
Step: 3
The number 22 is closer to 25 than 16.
Step: 4
So, the approximate value of 22 = 5.
Q6Estimate to the nearest integer.
- 79

A. -9
B. -7
C. -8
D. None of the above

Step: 1
79 lies between the perfect squares, 64 and 81. Step: 2
From the above number line, - 79 is nearer to - 81than to - 64.
Step: 3
So, - 79 is closer to - 9 than to - 8.
Step: 4
- 79 ≈ - 9
[Round to the nearest integer.]
Q7Estimate to the nearest integer.
130 + 23

A. 12
B. 13
C. 11
D. None of the above

Step: 1
130 + 23
[Original numerical expression.]
Step: 2
130 + 23 = 153
[Add the terms inside the square root.]
Step: 3
130 + 23 = 153
[Substitute in the expression.]
Step: 4
153 lies between the perfect squares, 144 and 169 as shown in the following number line. Step: 5
153 is nearer to 144 than to 169.
Step: 6
So, 12 is the nearest integer for the expression.
Q8The area of a square is 1345. Approximate the value of this.
A. 45
B. 37
C. 30
D. 40

Step: 1
Find the value of the 1345 using a calculator.
Step: 2
From the calculator, the value of 1345 is 36.674241.
Step: 3
In the decimal number, the digit in tenths place is 6.
Step: 4
As 6 is greater than 5, the decimal 36.6 is closer to 37 .
Step: 5
Therefore, the approximate value of 1345 is 37 .
Q9The length of the side of a square is 672. Approximate this length.
A. 30
B. 28
C. 26
D. 25

Step: 1
672 lies between the perfect squares 625 and 676.
Step: 2
672 is nearer to 676, than to 625.
Step: 3
So, the approximate value of 672 is 26.
Q10Approximate 174 to the nearest integer.

A. 12
B. 14
C. 16
D. 13

Step: 1
174 lies between the perfect squares 169 and 196.
Step: 2
174 is nearer to 169 than to 196.
Step: 3
Therefore, the approximate integer of 174 is 13.
Q11Approximate 32 to the nearest tenth

A. 5
B. 4
C. 6
D. 7

Step: 1
The perfect squares nearest to 32 are 25 and 36.
Step: 2
So, the value of 32 must lie between 5 and 6.
Step: 3
The number 32 is closer to 36 than 25.
Step: 4
So, the approximate value of 32 = 6.