PA_M6_S1_T1_Square Roots of Perfect Squares Transcript
Remember that when I did repeated multiplication, I was able to express
that with an exponent. If I have x squared, it's the same thing as x*x.
x2 is called the square of the number of x in this case. If a2 = b, then
a, the value that is being squared, is called the square root of b.
Some examples are down below with numbers.
92 is the same as 9*9, that's 81. The number that's
being squared is 9, so 9 is the square root of 81.
Here is another example, (-4)2. When I
do that, it's the same thing as (4)*(-4) and that's equal to positive
16, because two negatives make a
positive. In this case we find that -4
is a square root of 16. In fact,
positive 4 is also a square root of
16, and -9 is also a square root of 81, because when I multiply two of
the same sign together I get a positive result. So there's always the
possibility of a negative and a positive root whenever I take the square
root of a number.
Let's look at these examples and see if we can use what we
know about repeated multiplication to determine what the
square roots of these expressions are. In the first case
49 is equal to 7*7. That's 72. The number that's being
squared is 7, so 7 is a square root.
But, 49 is also equal to -7 * -7, so
that -7 is also a square root of 49.
Let's look at 196. Sometimes I need to
do a little bit more digging to find my
square root. In the case of 196 I am
going to write it as its prime factors.
The prime factors for 196 are 2*2*7*7. I
notice that I have two factors of 2 and
two factors of 7. That gives me two factors of 14. The
repeated factor, 14, is my square root, but I could
also get 196 by multiplying -14 times -14, -14 is also
a square root of 196.
Typically when we write these, we write the negative first because it's
the smaller value. When you give both roots for a square root, you write
the negative one first and the positive one second.
Let's look at these next two. 0. I want
root of 0. The definition of a square is
number squared that gives me my result.
square anything and get 0 back, is if a
root of 0 is 0. There's no negative and
associated with it, it's just 0.
to find the square
that I have some
The only way I can
= 0, so my square
no positive
Finally, let's look at the square root of -25. I want to
find a number that, when it's squared, gives me -25. But,
if I'm squaring a, it means I square whatever sign is
associated with a, as well. In other words, my repeated
multiplication is always of the same sign, meaning I
always get a positive result. So that there are no real
square roots, and that's true for any negative number.
Everything I've done here is an example of what we call perfect square
roots. It means I have a single integer that when I multiply it by
itself, gives me my squared term perfectly. In other words, in the case
of 196, if I divide 196 by 14 I get 14, and it's an even division
problem, there is no remainder left, same thing with 7 times 7, or -7
times -7, for 49. If I divide 49 by -7, I get -7 back.
It's an even division, there are no remainders left when I'm dealing with
the square roots of perfect squares.