Download The Mathematics 11 Competency Test

The Mathematics 11
Competency Test
Simplification of Square Roots by
Removal of Perfect Square
Factors
We can exploit the rules for multiplication of square roots described in the previous notes to
attempt to rewrite square roots in what is considered to be a simpler form. Here,
“simpler square roots”
means
“the number inside the square root is
smaller”
“Simplifying a square root” means rewriting it as an expression of the same value, but with the
number or expression inside the square root as small or simple as possible.
We will illustrate the technique here for square roots involving just numbers, but this method is
most important in simplifying square roots containing algebraic expressions.
As an example, notice that we can do the following:
3 5
 9  5
because 3 and
 95
by the rule for multiplying square roots together
9 represent the same number
 45
Thus
45 has the same value as 3 5 . But, we would consider 3 5 to be a simpler form
because the quantity in the square root is a smaller number. If we rewrite the above example
with the steps in reverse order, we can see the strategy for simplifying a square root when that is
possible.
45
If possible, separate or factor 45 into a product of two
numbers, one of which is the square of a whole
number. (Recall, we called such numbers “perfect
squares” earlier.)
 95
 32  5
 32  5
Use the rule for multiplying two square roots.
 3 5
since the square root of a square is the original
number.
3 5
The multiplication symbol can be omitted.
Since the remaining number in the square root, the 5, obviously cannot be written as a product of
a perfect square and another number, we have achieved as much simplification here as is
possible.
This strategy for simplifying square root expressions requires us to develop a strategy for
deducing how numbers can be rewritten as a product involving one or more perfect squares –
indeed, we need to be able to rewrite the original number in the square root as a product of
perfect squares, and the one smallest value which is not a perfect square. The easiest approach
for square roots involving relatively small numbers is just systematic trial: check to see whether
22 = 4, 32 = 9, 42 = 16, 52 = 25, etc. are factors of the number. We’ll illustrate this approach with a
few simple examples.
David W. Sabo (2003)
Simplifying Square Roots
Page 1 of 3
Example: Simplify
24 .
solution: We can write
24 = 4 x 6
The value 4 = 22 is a perfect square. The value 6 is not a perfect square, and cannot be written
as a product of a perfect square and some other number. Thus, in one step, we’ve achieved the
required factorization of the original value, 24. Thus
24  4  6
 22  6
 22  6
 2 6
2 6
Example: Simplify
50 .
solution: Going through our list of perfect squares systematically, we get:
try 22 = 4

try 32 = 9

try 42 = 16

try 52 = 25

50
4
50
9
50
16
50
25
 12.5 is not a whole number, so 4 does not go evenly into 50.
 5.55 so 9 does not go evenly into 50.
 3.1 so 16 does not go evenly into 50.
 2 , a whole number, so 25 is a perfect square factor of 50.
In fact, we can write
50 = 25 x 2 = 52 x 2
so that
50  52  2
 52  2
 5 2
5 2
Since 2 contains no whole number perfect square factors itself, this is as simply as we can write
50 .
David W. Sabo (2003)
Simplifying Square Roots
Page 2 of 3
Finally, one more example which requires quite a bit of work. Study the strategy used here to
achieve the complete factorization of the original number in the square root.
Example: Simplify
288
solution: We need to proceed fairly systematically in factoring 288 into a product of perfect
squares and at most a single factor which is not a perfect square because this is quite a large
number to work with.
Start by checking whether 4 is a factor of 288. Since 288/4 = 72, a whole number, we have that
288 = 4 x 72 = (22) x 72.
Now, check if 4 is a factor of 72. Since 72/4 = 18, a whole number, we have that
288 = (22) x 72 = (22) x 4 x 18 = (22) x (22) x 18.
Now check if 4 is a factor of 18. Since 18/4 = 4.5, is not a whole number, we cannot write 18 as a
product of 4 and another whole number. This means that there are no more factors of 4 to be
found in the original number.
So, now check if 9 is a factor of 18. Since 18/9 = 2 is a whole number, we have that
288 = (22) x (22) x 18 = (22) x (22) x 9 x 2 = (22) x (22) x (32) x 2
We have now expressed 288 as the product of three perfect squares and the value ‘2’, which
obviously cannot be expressed as a product of a perfect square and some other number. This is
a good place to take a minute and use a calculator to check that the factorization above really
does multiply out to give 288. Having quickly verified that we are error-free so far, we can now
proceed to simplify the square root:
288 
 2  2 3   2
2
2
 22
2
22
32
2
 223 2
 12 2
That is, in simplest form, we can write
288  12 2
NOTE: The same sort of methods can be used in principle to simplify higher order roots, except
one would have to look for factors which are perfect cubes or perfect fourth powers, etc. This sort
of problem is beyond the scope of the BCIT Mathematics 11 Competency Test.
David W. Sabo (2003)
Simplifying Square Roots
Page 3 of 3