In class, we worked on simplifying square roots, which means taking sqrt(n) and turning it into a x sqrt(b), where b is a square free number. We used what I call the prison break method, where a pair of twins under the square root sign can make a break for it, but only one gets out and the other is shot trying to escape. (Thanks, Mrs. Kruger!)

sqrt(20) = sqrt( 2 x 2 x 5), so one of the pair of 2s can be brought out, while the 5 stays inside.

sqrt(20) = 2 x sqrt(5)

We can also go to our calculator and get an approximation of sqrt(20) = 4.472135955..., a decimal number that continues on forever and never gets into an infinitely repeating pattern because it is an irrational number. If we round, we get an approximation of sqrt(20), but the idea is that sqrt(20) = 20 exactly, and the approximations won't be exactly 20 when we square them.

sqrt(20) rounded to one place after the decimal = 4.5

4.5² = 20.25

sqrt(20) rounded to two places after the decimal = 4.47

4.47² = 19.9809

sqrt(20) rounded to three places after the decimal = 4.472

4.472² = 19.998784

Approximating to three places after the decimal is fairly standard, since the square of that approximation is usually off by less than a hundredth. For example,

sqrt(999) = 31.60696126... or 31.607 approximated to three places after the decimal

31.607² = 999.002449

Practice problems.

For the following square roots, simplify and approximate to three places after the decimal, then find the square of the approximation.

sqrt(72)

sqrt(73)

sqrt(74)

sqrt(75)

Answers in the comments.

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For the following square roots, simplify and approximate to three places after the decimal, then find the square of the approximation.

sqrt(72)

72 = 2 x 2 x 2 x 3 x 3

sqrt(2 x 2 x 2 x 3 x 3) = 2 x 3 x sqrt(2) =

6 x sqrt(2)

sqrt(72) ~ 8.485

8.485² = 71.995225

sqrt(73)

73 is a prime, so sqrt(73) is as simple as it gets.

sqrt(73) ~ 8.544

8.544² = 72.999936

sqrt(74)

74 = 2 x 37, so it's a square free number and sqrt(74) is as simple as it gets.

sqrt(74) ~ 8.602

8.602² = 73.994404

sqrt(75)

75 = 3 x 5 x 5

sqrt(3 x 5 x 5) =

5 x sqrt(3)

sqrt(75) ~ 8.660

8.660² = 74.9956

Note: All the squares of the approximations are less that the actual number we are trying to find the square root for. This is just a coincidence that all the approximations rounded down instead of up when take to three places after the decimal. If the approximation rounds up, the square of the approximation will be more than the intended number.

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