# How To Estimate Square Roots

In order to estimate square roots, one must first list out a few perfect squares and their corresponding number: Numbers such as 1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, and 7^{2}, which correspond with 1, 4, 9, 16, 25, 36, and 49. Once you have these written out you can begin to estimate what the square root will be close to. For example, take the square root of 48, which is not easily done in mathematics. But using the numbers listed before we can find an estimate. 48 is one away from 49. So the answer to the square root of 48 would be around 7.

## Estimating Square Roots

In this problem here, we’re asked to estimate the decimal values for some square roots. Now before you can accurately estimate a square root, you need to know the values of the whole number’s squares, and I’ve supplied the first 13 of these over here on the right side of the board.

The process would be if we’re asked to find, for instance, the square root of 10 we would come over here and see which two numbers 10 is between, in this case its between 9 and 16, so we could say with certainty that because 10 is between 9 and 16, that the square root of 10 is going to be between 3 and 4, and since 10 is closer to 9 than it is to 16, we could also say that the square root of 10 is going to be closer to 3 than it is to 4.

That’s kind of what we’re going to be doing here, so our first problem is to estimate the square root of 40, so we’ll come over here and we can see that square root of 40 is going to be between 6 and 7, because 40 is between 36 and 49, so 40 is 4 more than 36, and there’s a difference of 13 between 49 and 36.

40 is about 4/13 of the way between, the square root of 40 is going to be about 4/13 of the way from 6 to 7. 4/13 is a little bit less than 1/3, so we’re going to estimate that this is about point 3, so we’re going from 6 to 7, so our estimate for the square root of 40 is going to be 6 point 3.

The second example we have 78, square root of 78, so we can look over here and say, “Okay, it’s going to be between 8 and 9.” 78 is 14 more than 64, and there’s a difference of 17 between 64 and 81, so the square root of 78 is going to be approximately 14/17 of the way from 8 to 9. 14/17 is a tiny bit less than 5/6, so we’re going to estimate this at point 8, so our estimate for this square root is going to be 8 point 8.

In this final one we have the square root of 132. The square root of 132 falls right along here, so it’s going to be between 11 and 12. 132 is 11 more than 121, and there’s a difference of 23 between 144 and 121, so the square root of 132 is going to be about 11/23 of the way from 11 to 12. 11/23 is just a tiny bit less than 1/2, so we’re going to estimate this decimal value at point 5, so our estimate for this square root is going to be 11 point 5.