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How to Figure Square Roots (Reducing)

Updated on March 19, 2015

How to Figure "Square Roots" of Numbers that are not Perfect Squares

I am a math tutor. I've been tutoring math since I was in high school back in the last century (well back in the last century). I began in 10th grade tutoring 9th graders in Algebra. The next year, I began tutoring Geometry (my favorite). I also got my first paying tutoring job that year (most of my high school tutoring was volunteer, after school, in the school building).

When I was in college, I tutored high school students in a program called "Upward Bound". I now tutor privately (in my own community) and through an on-line site called "Wyzant".

I bring this up because there are several topics I have noticed trouble my students and I thought this might help them (and other math students) with these subjects (I have other math related lenses).

This is not a lens about using a calculator to find a square root. This is not a lens about finding a square root to its nearest 10th or 100th or 2 places. This is a lens that helps you get from a large number's square root to a smaller number's square root.

In other words, this will show you how to go from ([square root] 20) to (2 ([square root] 5).

Reducing Irrational Square Roots

One of the sorts of problems that shows up on Algebra tests is taking a square root of a number that isn't a perfect square (see a list of all the integers that are perfect squares from 1-625 at the bottom of this lens) and reducing it to an easier to handle version. For example, in the picture here, we start with [square root] 98. But, if you notice when you look at the list, 98 is NOT a perfect square.

98, however, equals 2 X 49. 49 is a perfect square -- it's 7 squared (or 7 X 7). So, as you can see in the second step, [square root] 98 = [square root] 49 X [square root] 2. Or, more succinctly, 7 X [square root] 2.

If the number you are trying to divide is one you don't know (as in you can't tell right away by looking at it what perfect square might be a factor), check out Lowest Common Denominator to see how to break numbers into their factors. Or, if the number is low enough, you can look at the list of squares from 1-625 further down in this lens and work from there.

Perfect Squares

This is a list of the first 25 positive integers with their squares.

  1. 1
  2. 4
  3. 9
  4. 16
  5. 25
  6. 36
  7. 49
  8. 64
  9. 81
  10. 100
  11. 121
  12. 144
  13. 169
  14. 196
  15. 225
  16. 256
  17. 289
  18. 324
  19. 361
  20. 400
  21. 441
  22. 484
  23. 529
  24. 576
  25. 625

Compugraph Designs Arts Now Site

"Arts Now" is another "Print on Demand" site. They have a nice collection of clocks and watches, including the one pictured here (with math symbols on it). Click on the picture to see the entire math clock collection on the Arts Now site.

Did the help you understand square roots?

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    • srsddn lm profile image

      srsddn lm 5 years ago

      Yes, certainly. My son liked it very much