Approximating Square Roots


 
 
Concept Explanation
 

Approximating Square Roots

To find the approximate square root of a number, we shall use the method of long division. Using this method we shall obtain square root correct to certain decimal places. The steps are as followed

Step I: Obtain the number whose square root is to be computed.

Step II: Determine the number of decimal places to which the square root of the number is to be computed.

Suppose the square root of the given number is to be computed correct to n places of decimal.

Step III: Count the number of digits in the decimal part. If the number of digits is less than 2n, then affix a suitable number of zeroes at the extreme right of the decimal part so that the number of digits in the decimal part becomes 2n.

Step IV: Use the method of long division to find the square up to (n+1) places of decimal.

Step V: Check the digit at (n+1)th decimal place, if it is less than 5, then delete it to get the answer correct to n decimal places.If the digit at (n+1)th decimal place is 5 or more than, then increase the digit at nth decimal place by one and delete the digit at 9n+1)th place to obtain the square root correct up to n decimal places.

Illustration: Find the square root of 2 correct to three places of decimal.

Solution: Since we have to find the square root of 2 correct to three places of decimal,we shall first find the square root of 2 up to four places of decimal.for this purpose,we affix 8 zeros to the right of the decimal point.o we write

2 = 2.00000000

Now mark off periods and proceed as under:

therefore sqrt{2}=1.4142  up to four places of decimal.

Rightarrow sqrt{2}=1.4144 , correct up to three places of decimal.

Hence sqrt{2}=1.4144

Approximate values of Square Roots Using Square Root Tables

In many practical problems, we need the square roots of numbers, and finding approximate values of square roots of numbers by the method of long division is very time-consuming.for this reason, tables have been prepared which provide the approximate values of square roots of different numbers correct to a certain decimal place.

The following table gives values of square roots of all-natural numbers 1 to 24.

x sqrt{x} x sqrt{x} x sqrt{x} x sqrt{x}
1 1.000 7 2.646 13 3.606 19 4.359
2 1.414 8 2.828 14 3.742 20 4.472
3 1.732 9 3.000 15 3.873 21 4.583
4 2.000 10 3.162 16 4.000 22 4.690
5 2.236 11 3.317 17 4.123 23 4.796
6 2.449 12 3.464 18 4.243 24 4.899

Illustration: By using the table for square roots, find the value of sqrt{7}.

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Sample Questions
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Question : 1

By using the table for square roots, find the value of sqrt{frac{37}{64}}

Right Option : A
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Question : 2

What is the divisor in the third step while finding the square root of 66.

Right Option : B
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Explanation
Question : 3

Find the divisor in the fourth step while finding the square root of 2 using the approximation method.

Right Option : A
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Explanation
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