>> Welcome to an introduction to perfect squares and square roots.
In order to better understand square roots, we need to spend a little bit of time talking
about special numbers called perfect squares.
Perfect squares are the numbers that we obtain when we square an integer.
So here we see the integers from 1 through 10, and if we square them, we get perfect squares.
So if we wanted to we could expand them.
One squared equals 1 times 1, or 1.
Two squared is equal to 2 times 2, or 4.
Three squared would be 3 times 3, or 9...
16, 25. Over on the right we have 36, 49, 64, 81, and 100.
So all of these are considered perfect squares, and, again, if we take an integer
and square it, we obtain a perfect square.
Another way to think of a perfect square that will help when simplifying square roots is
to think of a perfect square as a number that has 2 equal integer factors.
So now that you have a feel for perfect squares, let's go ahead and talk about square roots.
To determine the square root of a number we have a special symbol as we see here.
This is the square root of 9.
The square root of a number is a number times itself that equals the given number.
What that means is, to determine the square root of 9,
we need to determine what number times itself would give us 9,
and since 9 is a perfect square, we can write 9 as 3 times 3,
or if we wanted to, we could write 3 squared.
Therefore the square root of 9 is equal to 3, and again, the reason it's 3 is
because 3 times 3 is equal to the given number 9.
Therefore the square root of 36, again, 36 is a perfect square because we can write
that as 6 times 6 or even 6 squared if we want, and this will equal one factor of 6.
Again, the reason this is 6 is because 6 times 6 is equal to the given number, 36.
The square root of 49, because that's 7 times 7, or 7 squared,
would be equal to 7, and let's look at one more.
The square root of 81, because 81 is 9 times 9, or 9 squared, would equal one factor of 9.
One more thing I'd like you to notice is that taking the square root of a number
and squaring a number are opposite operations or could be called inverse operations.
Notice here where we have the square root of 3 squared, the square root
and the square undo each other, and the result is just 3.
Here we have the square root of 6 squared.
The square root and the square undo each other, and the result is one factor of 6.
There's one more thing we need to discuss.
Actually numbers have two square roots.
If I asked you what are the square roots of 25, notice we could obtain 25 from the product
of positive 5 times positive 5 and also from the product of negative 5 and negative 5.
So if somebody asked what are the square roots of 25,
we would actually say positive 5 and negative 5.
However to avoid confusion, when we want the positive square root of a number,
we use this notation here called the principal square root,
and if we want the negative square root of a number, we just put a negative in front
of the square root, and this avoids confusion between which square root we're referring to.
Now let's go ahead and take a look at two more square roots.
If we take a look at the square root of 20,
there isn't a whole number times itself that would give us 20.
So square roots only simplify nicely when this number under here,
called the radicand, is a perfect square.
The closest perfect square that is less than 20 would be 16, and the closest perfect square
that is greater than 20 would be 25, and since the square root of 16 is equal to 4,
and the square root of 25 is equal to 5, the square root of 20 must fall somewhere between 4
and 5, and in order to get a closer approximation of the square root of 20,
we usually just use a calculator.
So let's go ahead and do that.
On this calculator if we press second x squared, it brings up the square root symbol,
and we just type in 20, a closed parenthesis, and then press enter.
So if we wanted to round this to the nearest hundredth,
we could say the square root of 20 is approximately 4.47.
Remember what that means is we take 4.47 and we square it, we should get approximately 20.
Notice it's slightly less than 20 because we did round this value.
So unless we're taking the square root of a perfect square,
we will normally have to get a decimal approximation.
Let's go ahead and try one more.
Sixty-one is not a perfect square.
The closest perfect square that is less than 61 would be 49, and the closest perfect square
that is greater than 61 would be 64.
Well, since the square root of 49 is equal to 7 and the square root of 64 is equal to 8,
the square root of 61 must fall somewhere between 7 and 8, and since 61 is closer
to 64, this value will be closer to 8.
Let's go ahead and use our calculator to get a decimal approximation of the square root of 61.
Again, second x squared, we type in 61, close parenthesis, press enter,
and it looks like it's approximately 7.81, which means if we take 7.81 and multiply it by itself,
[background ringing] we should get approximately 61.
Let's go ahead and try it, and, again you can see it's slightly less
than 61 because we did round.
Just to summarize this idea, if we're taking the square root
of a perfect square, it works out very nicely.
If we take the square root of a number that's not a perfect square,
we're only able to get a decimal approximation.
I hope you found this introduction helpful. ------------------------------66b95ce0be33--