How to estimate the square root of a number that is not a perfect square

1. How to estimate the square root of a number that is not a perfect square

2. square 9 3 • The square root of a number is a value that, when multiplied by itself, gives the number. • When finding the square root of a number the square root symbol is used to denote the principal (positive) square root. The square root symbol is called a radical and the number under the radical is called the radicand. • Essentially taking the square root of a number is the inverse operation or opposite of squaring a number. square root Principal Square root radical radicand

3. Any positive number has two square roots – a positive and a negative. (or = +36 (or = +36 So This is the same as

4. Now you try … What are the two square roots of 64? We know that (or = +64 (or = +64 So This is the same as

5. Important Note: When finding the square root of a number you are usually asked to find the principal square root or non-negative root of that number unless otherwise denoted.

6. You know to find the negative square root of a number when the negative sign is on the outside of the radical. Now you try ….

7. Be mindful though that the is not a real number. The reason is because there are not two identical real numbers that when multiplied together results in -25. 5 x 5 is positive 25 and -5 x -5 is positive 25. is considered “not a real number” and taking the square root of -25 results in an “imaginary” number. The explanation of “imaginary” numbers is beyond the scope of this podcast.

8. So it is easy to find the square root of a perfect square that is an integer but what about square roots such as , in which 30 (the radicand) is not a perfect square?

9. First, in order to understand how to estimate square roots of non-perfect squares, or a square root of a number that does not result in an integer, let’s first list the perfect squares and their roots.

10. Next, figure out which two integers that is between. = 6 We see thatis between the integers 5 and 6. This means thatis 5.?

11. But what does the exactly equal? • 30 is not a perfect square, so there are no two identical factors that when multiplied together will result in 30. • So when taking the it will not result in a an integer (whole number or the negative of a whole number). • However, we can drill down to find the two identical factors that when multiplied together will give you a number close to 30.   • Usually when estimating the square root of a number that is not a perfect square such as 30 without a calculator, estimating to the tenths or hundrethsdecimal place is sufficient.

12. Let’s try it.

13. We know that the square root of 30 is “5 point something”. Next, square 5.1, 5.2 and so forth and place it in a table. From the table we see that 5.42 results in 29.16 and 5.52 results in 30.25. So the square root of 30 falls between 5.4 and 5.5.

14. We can drill down even more to the hundrethsplace by creating another table. From here we can see that 5.472 and 5.482 results in a number closest to 30.

15. But which is the closest? We canfigure this out by subtracting. So for the square root of 30, estimating to the hundredths place we see that 5.48 is the best estimate because the difference is the smallest. We can repeat this process and drill down further to the thousandths, ten thousandths place and so on. We’d be here forever because there is no terminating or repeating decimal that when multiplied by itself results in 30. We can check our result of 5.48 on a calculator, and we get 5.477225575051661… 5.4772… rounds up to 5.48