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Chris Lele

Mental Math Shortcut: Perfect Squares – 1 | SAT Video Post

A Handy SAT Math Shortcut – N2 – 1

Many of us have a pretty good sense of what the squares of the first 15 integers are. Sure, you might be a bit shaky on ‘13’ and ‘14’ but you should be comfortable with the rest. Just to be sure, I’ll reproduce those below:

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100

11^2 = 121
12^2 = 144
13^2 = 169
14^2 = 196
15^2 = 225

Here are more squares students tend to know:
16^2 = 256
20^2 = 400
25^2 = 625
30^2 = 900

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(If you know all these, that’s pretty solid! You don’t have to memorize anymore.)

Why did I even bring this up in the first place? Well, I have a cool mental math shortcut. Assuming you know the above, you also know the following:

11 x 13, 14 x 16, 15×17 and even the crazy 29×31.

How is that possible?

Well, what if I told you that n^2 – 1 = (n – 1)(n + 1)

Big deal, you say. You already know basic algebra? And what does this have to do with squares?

Well, let’s say n = 20.

See, by knowing that 20^2 = 400, then the product of one integer less than 20—the number 19—and one integer greater than 20—the number 21—will be 400-1, which equals 399.

Try it with any of the numbers above. For instance, we know that 12^2 = 144. Therefore, 11 x 13 = 143.

29 x 31?

Well, what’s 30^2 – 1.

Just like that, voila! You’ve doubled your knowledge of squares above.

About Chris Lele

Chris Lele is the GRE and SAT Curriculum Manager (and vocabulary wizard) at Magoosh Online Test Prep. In his time at Magoosh, he has inspired countless students across the globe, turning what is otherwise a daunting experience into an opportunity for learning, growth, and fun. Some of his students have even gone on to get near perfect scores. Chris is also very popular on the internet. His GRE channel on YouTube has over 10 million views. You can read Chris's awesome blog posts on the Magoosh GRE blog and High School blog! You can follow him on Twitter and Facebook!

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