A common task in algebra is to simplify square
roots, or what is referred to in later mathematics
as radicals. This article will use the sqrt (x)
notation to mean "the square root of a number x".
Sometimes quite easy task to simplify, but sometimes
requires the help of a special formula, together
with your knowledge of perfect squares and factors.
For example, this would be the case for a radical,
such as sqrt (80).
It is therefore very important, because if a radical
is not simplified, it is typically considered to be
wrong, and you will either not receive or partial
credit for your answer to an exam. This article
shows you the simple steps to perform this task.
This article assumes that you are familiar with the
basic concepts of square and "square rooting." See
the Resource section for more information on these
1. It's easy to simplify a radical, which is a
perfect square, such as sqrt (81). We can either use
a calculator, or we can use our knowledge of perfect
squares to get a response at 9, page 9 ² equal 9. .
We have to remember that-9 is also a solution to the
problem, although it would be discarded in
connection with a geometry problem involving
lengths, or if we only were asked to give the
principal square root.
2. Simplifying a non-perfect square radical such as
sqrt (20) involves a bit more work. We could use a
calculator to get a long decimal approximation of
the answer, but it is not what is meant by
simplifying the radicals. What we are being asked to
do, in essence, is to break the radical one another
in such a way that we are left with the product of
an integer times the square root of a prime number.
3. To do this, it is important to know the specific
property of radicals is shown above. In the simplest
form is the equation tells us that we can split the
radical of a product in the product of the radicals.
If you want to apply the formula at sqrt (20)
example above, we would break 20 of factors of 4 and
5. . We have then sqrt (4 x 5), which can be divided
into sqrt (4) times sqrt (5). SQRT (4) is 2, so we
know our final simplified answer is 2 times sqrt
(5). It is the response that would be expected on an
exam. Note how we cannot break down sqrt (5), since
5 is a prime number, whose only factors are 1 and
4. Sometimes students asking if they could have
broken 20 in other factors, such as 2 and 10. the
answer is that we could, but we would be sqrt (2
times 10), which would break into sqrt (2) times
sqrt (10). Since none of them is a perfect square,
we will not end up with an integer component of our
response, which is what we need.
5. Let's get back to the example of sqrt (80) in the
introduction. 80 can be broken up into many factor
pairs, such as 2 and 40, 4 and 20, 8 and 10, etc.
What we have to look for is the largest perfect
square factor 80, and use it. 4 is a perfect square
factor on 80, but there is a bigger:. 16. This means
that we must use 16 and 5, as our factor pair. We
have now sqrt (16 times 5) = sqrt (16) times sqrt
(5) = 4 times sqrt (5), as is our answer.
6. In the above example, if we had used the 4 and
20, as our factor pair, we would have lots of extra
work to do. We would be sqrt (4) times sqrt (20). It
will be 2 times sqrt (20), but then we would have to
break down sqrt (20), as we did before. By using the
largest perfect square factor 16, we got our answer
in fewer steps.
7. One last example:. SQRT (200). There are many
factors, several of which are perfect squares. We
want the largest perfect square factor is 100. It
gives us the sqrt (100) times sqrt (2), which
represent 10 times sqrt (2).
8. Please note that we have no way of reducing the
square root of a number, which is either the primary
or the product of two primes. For example, we may
not simplify the sqrt (13). It is a prime number
with someone perfects square factors. We are just
going to have to leave our response is.
Another example would be sqrt (6). 6 is not prime.
We could break it up in sqrt (2) times sqrt (3), but
none of them is a perfect square, so it will not
simplify. We just wanted to let our response as sqrt
(6). It does not have any perfect square factor.
A last example is sqrt (77). 77 is not prime, since
it has other than 1 and itself, but the other
factors are both primes. Since it has no perfect
square factors, we just have to leave the answer
alone, and it is right to do so.
9. Algebra students must ensure that they are
comfortable with this process. It comes up quite
often in mathematics, and there is no reason to make
a problem perfectly, but then lose partial or full
credit just because you do not simplify your square