A square root of a number is a number that you can square to get it, that
is a number that you can multiply by itself to get the number. So 2 is a
square root of 4, because 22=4 and 3 is a square root of 9, because
32=9. (-2)2 is also 4 and (-3)2 is also
9. Numbers that have square roots always have two, a positive one and a
negative one. But the square root symbol means only the positive one, so
we can have one answer to our problem. Negative numbers don't have square
roots, because when you multiply numbers with like signs you get positive
numbers.
What about a number like 2 or 5 or 10. Do they have square roots? Are there
numbers that you can multiply by themselves to get numbers like these? Clearly
there are no whole numbers that will work, but what about something involving
fractions? It turns out that fractions won't work either. It turns out that
if you can't find a whole number to square and get a given whole number,
no fraction will work either. But it also turns out that square roots of
all whole numbers do correspond to lengths. This is kind of strange I think,
lengths that can't be measured by whole numbers or fractions, but it really
is true. To measure such lengths mathematicians use irrational numbers.
Read my article Irrational Numbers
to learn more about this. You can also find out more about irrational numbers
by going on my pi tour.
You can approximate such square roots by rational numbers, and you can
get as close as you want, and it is quite easy to do so. The simplest way
is to just do a lot of guessing and checking. First find the two whole numbers
that it is between and then the nearest 10th, the nearest 100th, etc., etc..
For example if you want to find out what the square root of 2 is, you know
that it must be between 1 and 2, because 12=1 and 22=4
and 2 is between 1 and 4. 2 is closer to 1 than to 4, so the square root
of 2 must be closer to 1 than to 3, but not that much, so we might guess
1.3 or 1.4. If we guess 1.3 and square it we get 1.32=1.69, which
is much too small, so try 1.4. 1.42=1.96, still too small, but
1.52=1.25, which is much too big, so to the nearest 10th the
square root of 2 is 1.4. To get another decimal space since 1.42=1.96
is much closer than 1.52=2.25, we might try 1.41. 1.412=1.9881,
still too small. 1.422=2.0164, too big. 1.412 is closer,
so to the nearest 100th the square root of 2 is 1.41. Continuing like this
we should be able to get as close as we want. There are also fancier methods
that do it a bit quicker and your calculator uses one of these, and by using
a calculator you don't actually have to go through such a long process to
find a square root. But it is useful to try it at least once, just to make
sure you really know what a square root is. After that you can find an approximation
for any square root just by keying the number into the calculator and pushing
the square root button.
But since these will only be approximations anyway, most of the time in
mathematics we just leave the square root undone and use
as the name for the exact real number that you can square and get 2 in the
same way that we use 1/3 to designate the number you get when you divide
1 by 3 and don't always divide it out.
One thing that is important to do is to get using to using radical notation.
That funny symbol over the 2 is called a radical. When it is put around
a number the whole thing then means the positive number that can be square
to get that number. That means, for example, that 
means the positive number that you can square and get 4, which means that
it is equal to 2. Similarly an expression like
means we are adding the number we can square to get 4 to the number we can
square to get 9. The number we can square to get 4 is 2 and the number we
can square to get 9 is 3, so this would simplify to
2+3=5,
by simply replacing things with what they are equal to. One thing never
do is leave the radical there after you have done it. For example
would mean that the number you square to get 4 is the same as the number
you square to get 2, which is total nonsense. That is why I have crossed
it out in red, because I want to make sure you realize that this is wrong.
The radical symbol shouldn't be used like the long division symbol. The
long division symbol stays around when the division has already been done,
because it is not so much a symbol for the operation as a computational
tool for keeping track of the division similar to the line separating things
that you are adding, subtracting, or multiplying from the answer. In this
sense the radical symbol is more like the ÷ symbol, so saying something
like the above would be sort of like saying 14÷2=7÷2.
Whenever you have an expression with radicals in it and they can be evaluated,
all you have to do to simplify the expression is replace the radicals with
what they are equal to. So for example if you have
the square root of 9 is 3, so we replace it with 3, and the square root
of 25 is 5, so we replace that with 5, and we get
(5)(3)+(2)(5).
Then we do the rest of the arithmetic and get
15+10=25.
On the other hand, the radical symbol creates an automatic grouping. Any
operation inside it is done before taking the square root. So in
you don't find the square root of 4 and the square root of 9 and add them
together. Instead it means that you add the 4 and the 9 first, and get 13
and then try to find a number that you can square to get 13. In this case
since there is no rational number that you can do that with, the answer
would be just
.
The above kind of simplifying just depends on knowing what the notation
means, but there is some additional simplifying of radical expressions that
you can do by using a couple of properties. Here are the properties that
you need to know for this.
The first of these is helpful for writing irrational radicals with as small
as possible numbers left inside the radical. The idea is that if you have
something like
even though you can only approximate the answer to it with a rational number,
you can evaluate part of it and make the irrational part smaller, and it
is considered to be a simpler way to write it. What you do is you use the
first of the properties above to write it as the square root of 4 times
the square root of 3, and the important thing here is that the square root
of 4 can be evaluated.
Again just like before all I am doing here after I apply the property is
replacing things with what they are equal to. When you do this it doesn't
do to factor it any way. It is important to factor it so that one of the
factors is a perfect square so that you can evaluate it. Sometimes when
doing this if you have trouble finding a factor that is a perfect square
a good method is to prime factor the number. Then all of the pairs will
be perfect squares.
In the first radical here you are asking yourself what can you multiply
by itself to get 2 times 2, and of course the answer is 2. In the second
radical you are asking yourself what can you multiply by itself to get 3
times 3, and of course it is 3. In the third radical you are asking what
can you multiply by itself to get 3, and the answer isn't so obvious, because
it is a nasty irrational number, so it is best to leave it undone and use
the square root of 3 as a symbol for it. Again this is replacing things
with what they are equal to. In the last step you multiply the 2 and the
3 to get the final answer.
The second of these properties is useful for evaluating square roots of
fractions, because it allows you to do it by finding the square roots of
the numerator and the denominator individually.
You could also do this problem by asking yourself what can you multiply
by itself to get 4/9 and thinking about how fractions multiply, but it makes
it a little easier this way.
It is really important when using these properties to realize that these
properties are special to multiplication and division. The reason they work
has to do with the fact that powers are repeated multiplication. Powers
aren't repeated addition, so there are no similar properties for addition
and subtraction. The expression
means you are trying to find the number you can square to get the answer
to 9+16. That means you are trying to find the number you can square to
get 25. That is 5. On the other hand the expression
means you are find the number you can square get 9 and the number you can
square to get 16 and you are then adding them together. The number you can
square to get 9 is 3. The number you can square to get 16 is 4. The answer
then is 3+4=7. 5 and 7 are not equal.
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