Find Perfect Squares Mentally with this Trick
mental math series, part 6
Mental math tricks. Aren’t they great? Little cheats and shortcuts to get you to your destination quicker and easier. And I’ve got another for you! This one will enable you to calculate any two-digit perfect square mentally.
We talked about perfect squares once before in Lesson 2: The Decimal System, Exponents and a few Perfect Numbers. Check it out if you need a refresher!
At this point, you should be comfortable with perfect squares up to twelve squared.
And if you’re an over-achiever, you may even know up to sixteen squared. (✋Guilty.)
Which is awesome, but what if you could simply calculate these two-digit squares with one simple method?
Ready for this nifty trick?
The Squares Trick
Let’s solve fifteen squared using the trick.
Step one: Determine the distance 15 is away from 10, which is 5. Therefore add and subtract 5 from 15.
Step 2: Multiply the results together.
Step 3: Take the 5, square it and add it to 200.
Here’s a diagram reviewing the process altogether.
Example 2
Let’s try another example.
Begin by adding and subtracting 6 from 16.
Then multiply the results.
And add six squared to it.
Hence the answer to sixteen squared is 256. Again, here’s an overview.
Example 3
Now for a challenging one.
In this problem, add and subtract 3 since it gets us to the benchmark 80.
Because 80 x 86 is a large product, I split 86 into 80 + 6 in order to distribute the 80 through.
(Remember that when you multiply by multiples of 10, simply multiply the non-zero digits together and append an equal number of zeros to the result.)
After distributing the 80 through, I have the following:
Hence the answer is:
Why This Trick Works
By now you might be wondering why the heck is this working? Let’s walk through the theory of why this works, using fifteen squared as an example.
Recall fifteen squared is equivalent to:
To make the multiplication easier, we subtract 5 to get 15 down to the benchmark number 10. To account for taking away 5, add 5 to the other 15.
At this point, we have subtracted 25 from our product. To see this multiply the two terms together.
We’ll use a method called FOIL to multiply the terms, which stands for First, Outside, Inside, Last.
First
First refers to multiplying the first terms together from each binomial.
Outside
Outside refers to multiplying together the very first and very last terms.
Inside
Inside refers to multiplying the second and third terms.
Last
Last refers to multiplying the last terms in both parenthesis.
Combine the products together.
Because + 75 – 75 = 0, we have:
We can conclude that our adding and subtracting 5 has resulted in a deficit of 25. This is why we must add five squared to the equation.
Note: The criss cross multiplication trick from the last lesson works on two-digit perfect squares as well, if you prefer!
Next Lesson: Negative Exponents and the Decimal System
