How to estimate the square root of a number that is not a perfect square When you are ready to record start on slide 3! When you save your podcast as a video file, it will add these two slides automatically. Cool , huh?

3 9 9 =3 radical radicand square square root 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 3 9 square root Principal Square root radical radicand 9 =3

Any positive number has two square roots – a positive and a negative Any positive number has two square roots – a positive and a negative. (+6)·(+6) or (+6) 2 = +36 (−6)·(−6) or (−6) 2 = +36 So 36 = +6 𝑎𝑛𝑑 −6 This is the same as ±6

Now you try … What are the two square roots of 64? We know that (+8)·(+8) or (+8) 2 = +64 (−8)·(−8) or (−8) 2 = +64 So 64 = +8 𝑎𝑛𝑑 −8 This is the same as ±8

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.

You know to find the negative square root of a number when the negative sign is on the outside of the radical. − 49 =−7 Now you try …. − 81 =−9

Be mindful though that the −25 is not a real number Be mindful though that the −25 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. −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.

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

After taking the square root the result is 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. Number Squared 12 22 32 42 52 62 72 82 92 102 112 122 132 142 152 Perfect Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 After taking the square root the result is 2 3 5 6 7 8 10 11 13 14 15

Next, figure out which two integers that 30 is between. 25 =5 30 = ? 36 = 6 We see that 30 is between the integers 5 and 6. This means that 30 is 5.?

But what does the 30 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 30 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 hundreths decimal place is sufficient.

Let’s try it.

So the square root of 30 falls between 5.4 and 5.5. 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. Numbered squared Result 5.12 26.01 5.22 27.04 5.32 28.09 5.42 29.16 5.52 30.25 5.62 31.36 5.72 32.49 5.82 33.64 5.92 34.81 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.

We can drill down even more to the hundreths place by creating another table. Number Squared Result 5.412 29.2681 5.422 29.3764 5.432 29.4849 5.442 29.5936 5.452 29.7025 5.462 29.8116 5.472 29.9209 5.482 30.0304 5.492 30.1401 From here we can see that 5.472 and 5.482 results in a number closest to 30.

But which is the closest? We can figure this out by subtracting. 5.472 29.9209 30 – 29.9209 .0791 5.482 30.0304 30.0304 - 30 .0304 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