Question

I have doubt in this chapter

How to find non perfect squares without actual calculation

Answer 1

There are a number of ways to calculate square roots without a calculator. Here is guess and divide method.

Step 1: Estimate/find a perfect square root as close as possible to your number.

Step 2: Divide your number by the square root.

Step 3: Calculate the average of the result of step 2 and the root.

Step 4: Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

Calculate the square root of 10 to 4 decimal places.

1. Find the perfect square number closer to 10.

3²= 9 and 4² = 16,
so take 3.

2. Divide 10 by 3.
10÷3 = 3.33 (you can round off the answer)

3. Average 3.33 and 3. (3.33 + 3)÷2 = 3.1667

Repeat step 2:
10÷3.1667 = 3.1579

Repeat step 3: Average 3.1579 and 3.1667.

(3.1579 + 3.1667)÷2
= 3.1623

Try the answer --> Is 3.1623 squared equal to 10?
3.1623 x 3.1623
= 10.0001

example, √695 =?

1. Let us first guess for the root as 25.
2. 695 ÷ 25 = 27.8
3. (25 + 27.8) ÷ 2 = 26.4
4. 695 ÷ 26.4 = 26.3257
5. (26.4 + 26.3257) ÷ 2 = 26.36285
6. 695 ÷ 26.36285 = 26.3628553

Last two findings are about equal, so you now have a good estimate of the root of 695. You can keep going through the process until you are satisfied with the accuracy.

So, √24.6=?

1. Let's try 5 since 5² = 25, which is pretty close to 24.6.
2. Divide 24.6 by 5. 24.6 ÷ 5 = 4.92
3. (5 + 4.92) ÷ 2 = 4.96

You can stop here. 4.96 is pretty close to 4.9598 which is the actual square root of 24.6. Repeat steps 2 and 3 to any desired level of accuracy. The further you go, the harder the long division becomes. But the first few cycles yield a pretty close answer.

• 24.6 / 4.96 = 4.9596
• (4.96 + 4.9596) ÷ 2 = 4.9598

Some more examples solved concisely,

√2613 =?

√2500 = 50

2613 ÷ 50 = 52.26

(52.26 + 50) ÷ 2 = 51.13 (approx. answer)

√6673 =?

√6400 = 80

6673 ÷ 80 = 83.41

(83.41 + 80) ÷ 2 = 81.70