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Mental Math: Squaring Shortcuts
Now we’ll talk about some shortcuts you can use when you have to square numbers. First of all, it’s very easy to square a multiple of 10. If we need, for example, 40 squared, we’d begin by squaring the non-zero part. Of course, 4 squared is 16. Now put this in front of two 0’s, 1600, that’s it. That’s how you square a number divisible by 10.
40 squared is 1600. In this pattern, let’s think about this. 2 times 2 is 4, so 20 squared is 400. 3 times 3 is 9, so 30 squared is 900. 4 times 4 is 16, so 40 squared is 1600. 5 times 5 is 25, so 50 squared is 2500.
6 times 6 is 36, so 60 squared is 3600. 7 times 7 is 49, so 70 squared is 4900. 8 times 8 is 64, so 80 squared is 6400. 9 times 9 is 81, so 90 squared is 8100. And, finally, 100 squared is 10,000. Two zeros times two zeros gives us four zeros.
That’s how you multiply powers of ten. So, all of these numbers should be very easy to find, and you should not need a calculator. You should be able to find these in your head very quickly. Now, it’s slightly trickier to square the numbers that end in 5, the mul, the odd multiples of 5.
First of all, when we square any number ending in 5, the squared number ends in 25. That’s always what will form the rightmost two digits. So that’s important. So it’s just a matter of finding the stuff that goes in front of 25. Now think about 35 squared.
Since 35 is between 30 and 40, it must be true that 35 squared is between 30 squared and 40 squared. So in other words, 35 squared would have to be between 900 and 1600. So think about the digits. What digits are in the hundred place in front of the 25? Well, 3 times 3 is 9, and that’s too small.
4 times 4 is 16, that’s too big. Instead of 3 times 3 or 4 times 4, we’re gonna use 3 times 4. Which of course is gonna have to be between 3 times 3 and 4 times 4. That’s the number we put in front of 25. So what we get is 35 squared is 1225. And that’s the answer.
This typifies the general pattern. Any number endi, ending in 5 is always halfway between two multiples of 10. The product of the number in front of the zeros in these two multiples of 10 is crucial for figuring out the square of the number ending in 5. So the procedure to square a number ending in 5. Remove the 5 and leave the remaining digits, so that’s 7.
And then we’re gonna add 1 to it. Now one way to think about this, 75 is between 70 and 80. So those are the two numbers we need. Drop the zeros. We need the 7 and the 8. We’re gonna multiply those two numbers.
7 times 8 is 56. And again, you need to know your, your one-digit multiplication table well. 7 times 8 is 56. That’s the number we’re gonna put in front of 25. So 75 squared is 5625. It’s not often that we need to do this on the test, but when needed, this is a very handy trick.
So pause the video and do these practice problems without a calculator. Do these practice problems using mental math. Okay. 25 squared. Well, 25 is between 20 and 30, so we need the 2 and the 3. We multiply them together. We get six.
Put that in front of 25. We get 625. 25 squared is 625. Incidentally, because 25 is 5 times 5, that means that 5 times 5 times 5 times 5, 5 to the 4th is 625. 65, well 65 is between 60 and 70.
So we need the 6 and the 7. Those are the two numbers we need. When we multiply them together, we get 42. Put that in front of 35. We get 4225. That’s the answer.
Now 115 squared, well 115 is between 110 and 120. Take off the zeros. Those are the numbers 11 and 12. So we need 11 times 12. We don’t need a calculator for that. 11 times 12.
Well 11 is 10 plus 1. Well clearly 12 times 10 is 120. 12 times 1 is 12. So we just add those two, and we get 132. That’s the number we put in front of 25. So we get 1-3-2-2-5 or in other words 13,225.
That is 115 squared. We don’t need a calculator for that. That’s, that’s very straightforward. Now here’s an even more widely applicable trick. Think about any two adjacent squares. Say, 7 squared, which is 49, and 8 squared, which is 64.
The number 49 has seven 7’s in it. Suppose we add 7 to that. We get 56. 56 is eight 7’s, or seven 8’s. Because that’s two ways to think, the fact that, that 7 times 8 is 56, we can think of it as seven 8’s or eight 7’s.
Well, because it’s seven 8’s, if we want eight 8’s. All we’d have to do is add eight and then we’d get 64. So in other words, we start with 7 squared, we add 7, we add 8 and that brings us to 8 squared. And that generalizes to a very powerful pattern. If we know the square of any value, if we know n squared equals some number, then we can add that value n, then add the next integer up, n plus 1, and this will result in the next square up.
Imagine that we had n squared tiles in an n by n array. So you have to imagine we have the dot dot dots here, but we have this. Big n by n square tiles. So there’s n squared tiles in this total array. And we want to build this up to an n plus 1 times n plus 1 array. Well, how would we do that?
First of all, we’d add a column of n more tiles on the side, and this would produce a rectangle. So this, this rectangle is n squares high and it’s n plus 1 columns across. We have that extra one that we added. And so we added n by adding that column. Well, now we’re gonna add a top row, n plus 1 across the top and that will complete it and we’ll have a new square.
So what we’ve done is, we had n by n, we had n squared. We added n in that column, and then we added n plus 1 at the top, and that brought us to an n plus 1 times n plus 1 square. So in other words, n squared plus n plus n plus 1 equals n plus 1 squared. That is an incredibly powerful pattern. So we’ve already talked about how it’s easy to find the squares of multiples of 10, and there’s a good trick also for finding squares of numbers that end in 5, and this allows us very easily to find other numbers that are adjacent to those numbers.
Suppose we had to find, say, 41 squared. Well, of course, 40 squared is 1600. That’s very easy to find. Once we have that, well, 41 squared would have to be 40 squared plus 40 plus 41. So that would be 1600 plus 40 plus 41. That would be 1681.
Very easy to do, even without a calculator. Suppose we needed 69 squared. Well, we know 70 squared is 4900. Now we have to go down, we have to subtract 70 and then subtract 69. So 4900 minus 70. That would be 4830.
And then to subtract 69, what I would want to do with mental math, I’d want to just subtract 70 again and add 1. Subtract 70, add 1, we wind up with 4761. And that is 69 squared. So practice these without a calculator and then we’ll talk about them. Okay.
39 squared. Well of course, 39 squared is very close to 40 squared. 40 squared is 1600. We’re gonna start at 40 squared. We’re gonna subtract 40, then subtract 39, and that will bring us down to 1521.
1521 is 39 squared. 81 squared. Well, we know 80 squared is 6400. So from there we’re just gonna add 80, and add 81. And that brings us up to 6561, which is 81 squared.
56 squared. Well, we know 55 squared so, think about this. 55 squared is, 55 is between 50 and 60. That’s 5 and 6. 5 times 6 is 30. Put that in front of 25.
55 squared has to be 3025. And to this we’re gonna add 55 and 56. Well, 25 plus 55 is 80, so that’s 380, plus 56, so we take 20 more, that will bring us to 3100, and we’ll be left with 36. So it will be 3136. And that is 56 squared.
84 squared. Well, first we need to find 85 squared. 85 is between 80 and 90. Drop the 0, so that’s 8 and 9. 8 times 9 is 72. Put that in front of 25.
7225.That’s 85 squared. From there, we’re gonna subtract 85 then subtract 84. Okay. It’s a bit tricky. I’ll start by subtracting the 5s, then subtract the 8.
22 minus 8 is 14. So this brings us down to 740 minus 84. So I’ll subtract the 40, and then that will bring us down to 7100, and I’ll subtract the 44. That will bring us down to 7056. So, little bit of intense mental math there, but as you practice, you can get good at doing these calculations.
To square a multiple of 10, square the digits without the 0 and tack on two 0’s at the end. That’s the real easy trick. To square a number ending in 5. First we’re going to remove the 5, remain, leaving the remaining digits, that’s the multiple of 10 lower than it.
Add 1, that’s the multiple of ten higher than it. Multiply those two numbers. And then put that product in front of 25. And finally, the really big trick, if we know the value of n squared, and especially this is easy if n happens to be a multiple of 10 or multiple of 5, then we can get the next square up, which is n plus 1 squared, by adding n or adding n plus 1.
Or we could go down by 1 by subtracting n, and then subtracting the next one down. So these are very important mental math tricks when you need to find squares.
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