At some point square
roots should no longer be considered an operation but rather the most efficient
way to express a number. For example,
the best way to write one hundred trillion is . The best way
to express the number times itself that is two is as
That provides insight
when we consider multiplying a rational number and an irrational number
together. It is not confusing for some
irrational numbers, like π. Nobody
confused 3π because we understand that symbol is the best way to write the
number. There’s not a way to rewrite multiples of π other than by writing the
multiple in front.
However, is often
written as . There are
reasons explained by the order of operations which tell us why this is false,
but understanding what the square root of two is perhaps offers the simplest
The square root of six is approximately 2.449. Not the same thing at all.
The following, however, is true:
generalization can be used. Sometimes it
is best to write things one way versus another, and it is up to you to decide
if rewriting an expression offers insight.
If two numbers are both
square roots you can multiply their radicands together. But you cannot multiply
the radicand of a square root with rational number like we saw above.
Division is a little more
nuanced, but only when your denominator is a fraction.
This generalization is
true for division:
This can be calculated two ways.
But you cannot divide
rational numbers into the radicand, or the radicand of a square root into a
rational number. Remember, square roots,
when simplified, are the most efficient way of writing irrational numbers. If we used k to represent the square root of two, these types of confusing
things would not be happening.
Nobody would confuse what is happening with . We simply
cannot evaluate that because 6 and k
do not have common factors. When k is written as the square root of two,
sometimes people just see a 2 and reduce.
The only issue with
division of square roots occurs if you end up with a square root in the
Denominators must be
rational and the square root of two is irrational. However, there’s an easy
fix. Remember that and It is also true
To Rationalize the Denominator, which means make the denominator a
rational number, we just multiply as follows:
Sometimes we end up with
something like this:
Three is a rational
number and is perfectly okay in the denominator. If you multiply by the fraction you can still
get the simplified equivalent, but you’ll have extra reducing to do at the
end. Instead, just multiply by the
In summary, to divide or multiply
with square roots, you can multiply or divide the radicands. However, if you’re multiplying or dividing
rational numbers and square roots, you cannot combine the radicands and the
In this section we will
see why we can add things like but cannot add
things like . Later we will
see how multiplication and division work when radicals (square roots and such)
Addition and Subtraction: Addition
is just repeated counting. The
expression means , and the expression So if we add those two expressions, we get . Subtraction works the same way.
Consider the expression . This means The square
root of five and the square root of three are different things, so the simplest
we can write that sum is .
A common way to describe
when square roots can or cannot be added (or subtracted) is, “If the radicands
are the same you add/subtract the number in front.” This is not a bad rule of thumb, but it
treats square roots as something other than numbers.
The above statement is true. Five groups of three and four groups of three
is nine groups of three.
The above statement is also true because five groups
of the numbers squared that is three, plus four more groups of the same number
would be nine groups of that number.
However, the following
cannot be combined in such a fashion.
While this can be calculated, we cannot add the two
terms together because the first portion is three eights and the
second is five twos.
The same situation is happening here.
The following is
obviously wrong. A student learning this
level of math would be highly unlikely to make such a mistake.
Seven twos and nine twos makes a
total of sixteen twos, not
sixteen fours. You’re adding the number of twos you have together,
not the twos themselves. And yet, this
is a common thing done with square roots.
This is incorrect for the
same reason. The thing you are counting
does not change by counting it.
Explanation: Why can you add ? Is that a
violation of the order of operations (PEMDAS)?
Clearly, the five and square root of two are multiplying, as are the
three and the square root of two. Why
does this work?
Multiplication is a
short-cut for repeated addition of one particular number. Since both terms are repeatedly adding the
same thing, we can combine them.
But if the things we are
repeatedly adding are not the same, we cannot add them together before multiplying.
What About Something Like This: ?
Before claiming that this
expression cannot be simplified you must make sure the square roots are fully
simplified. It turns out that both of
these can be simplified.
The dot symbol for
multiplication is written here to remind us that all of these numbers are being
What About Something Like This: versus
Notice that in the first
expression there is a group, the radical symbol groups the sevens
together. Since the operation is adding,
Since the square root of
fourteen cannot be simplified, we are done.
The other expression
Summary: If the radicals are the same number, the number in
front just describes how many of them there are. You can combine (add/subtract) them if they
are the same number. You are finished
when you have combined all of the like
terms together and all square roots are simplified.
Perform the indicated