When it comes to multiplying squares together, multiplying small numbers like 4 times 4 or
8 times 8 is pretty easy, and in fact you've probably got most of those values memorized.
But when it comes to multiplying larger squares together, when we get up into the teens like
17 times 17, or certainly higher, maybe 33 times 33, multiplying squares starts to get
much more difficult, and we probably don't have any way to find the answer quickly without
a calculator.
And that's why I wanted to show you a trick for multiplying larger square numbers together.
Now the squares have to be equal to one another, so like I said we have to have 33 times 33
or 17 times 17.
But in a problem like that, there is a quick way to find the product.
So let's take, for example, 27 times 27.
We want to do this multiplication and we don't have a calculator.
So what's the trick for figuring this out?
Well, we'll follow the same process every time, it's always going to work when we multiply
squares like this.
And what we want to do is we want to take the first number, 27, and we want to round
it to the nearest 10.
So 27 is closer to 30 than it is to 20, so we're going to round this up to 30.
The difference between 27 and 30 is 3, and we're going to be coming back to that in a
little bit, so we want to make note of what the difference is.
Since we changed 27 by 3 to get up to 30, we then want to change the other 27, but we
want to move it down by 3.
So 27 minus 3 would get us down to 24.
So however much you move the first number down, you want to move the second number that
same distance, but in the opposite direction.
So 27 went up by 3 to 30, which means 27 has to go down by 3 to 24.
And that difference is 3 as well.
The differences will always be the same, because that's the rule that we're using, we're keeping
these changes the same.
So up by 3 and down by 3.
Then 30 times 24 becomes our new multiplication problem.
Now you might be asking, how is 30 times 24 any easier of a multiplication problem than
27 times 27.
Well, it actually is a lot easier, because when we have a multiple of 10, 30 is a multiple
of 10, so 10, 20, 30, 40.
When you have that, you can ignore the 0 here.
So this problem, 30 times 24, really you can think of as 3 times 24, ignoring that 0.
And 3 times 24 is certainly much easier, that's 72.
So 3 times 24 is 72.
And then that 0 that we ignored, we add back on to the end.
So 30 times 24 is 720.
So this actually is a much easier multiplication problem than 27 times 27.
And in fact, this gets us very close to our final answer.
All we have to do now is account for these 3's that we took out before.
So we take the 3s and what we want to do is multiply them together.
3 times 3 is going to be 3^2, or 9.
So this value here is what we want to add to the 720 that we got.
So 720+9, gives us 729.
And 729 is actually the value of 27 times 27.
So let's then look at this trick with another example.
How about 16 times 16.
And you might already have this value memorized, but let's say that you don't and you want
to figure it out quickly.
Well, we look at the first 16 and we follow the same process that we did in the last problem.
Is 16 closer to 10 or 20?
It's closer to 20 so we want to round this up to 20, which is a difference of 4.
That means we want to change the other 16 by 4 as well in the other direction.
So 16 plus 4 to 20, in this case 16 minus 4 down to 12.
And that was a change of 4 as well.
Now 20 times 12 is our new multiplication problem, which is much easier because we can
ignore the 0 here, and this is really just 2 times 12.
2 times 12 is 24, and then we just pop our 0 back on the end.
So 20 times 12 is 240.
Now we're close to our final answer, except that we have to bring these 4's back into
play, so again we multiply them, 4 times 4 is 16.
And then we add 16 back to 240, and we get 256, and 256 is the product of 16 times 16.
And that's the trick you can use for multiplying larger squares together.