MAT 1275: Introduction to
Mathematical Analysis
Dr. A. Rozenblyum
II. Exponents and Radicals
D. Products and Quotients Involving Radicals
Products and Quotients Involving Radicals
Simplification of products of expressions that contain radicals is based on the following
multiplication rule that we mentioned in section II.B:
n
a  n b  n ab .
This rule allows to combine a product of two (or more) radicals in one radical.
Example 1. Multiply and simplify 5 7  3 14 .
Solution. We can multiply separately numbers outside the radicals (5  3) and numbers
inside the radicals (7 14) . To do this, it may be convenient to re-arrange numbers in
given expression to group together numbers outside and inside the radicals:
5 7  3 14  5  3  7  14
=15 7 14  15 7  7  2
=15  7 2  105 2 .


Example 2. Multiply and simplify 3 5 2 15  4 30 .
Solution. We can open parentheses using the usual distributive property:


3 5 2 15  4 30  3 5  2 15  3 5  4 30 .
Next, we can process each item separately as in Example 1:
3 5  2 15  3  2  5  15  6 5 15  6 5  5  3  6  5 3  30 3 ,
3 5  4 30  3  4 5  30  12 5  30  12 5  5  6  12  5 6  60 6 .
From here


3 5 2 15  4 30  30 3  60 6 .
Note. As you can see, the final answer contains two radicals. We cannot combine them in
one radical because they have different numbers inside (that cannot be processed further).
Products and Quotients Involving Radicals



Example 3. Multiply and simplify 3 2  4 5 3  6 .
Solution. As in Example 2, we start with distributive property:
3


2  4 5 3  6  3 2 5 3  3 2  6  45 3  4 6 .
Again, we can process each item separately (if you want, you may process them
simultaneously):
3 2  5 3  15 6 ,
3 2  6  3 12  3 4  3  3  2 3  6 3 .
4  5 3  20 3 .
From here,
3


2  4 5 3  6  15 6  6 3  20 3  4 6
=15 6  6 3  20 3  4 6  (15  4) 6  (6  20) 3  11 6  14 3 .
Products and Quotients Involving Radicals
Bellow, we consider an example that can be easily solved using the following simple but
useful formula
( a  b )( a  b )  a 2  b 2 .
You can check this formula by opening parentheses on the left.



Example 4. Multiply and simplify 2 5  3 7 2 5  3 7 .
Solution. We can solve this problem in the same way as in Example 3, using the
distributive property. Notice, however, that in both pairs of parentheses we have
difference and sum of the same expressions. Therefore, we can use the above formula
with a  2 5 and b  3 7 . We have
 
 5   4  5  20 ,
 3 7   3   7   9  7  63 .
2
a 2  2 5  22 
b2
From here,
2
2
2

2
2
    
2
5 3 7 2 5 3 7  2 5  3 7
2
 20  63  43 .
Products and Quotients Involving Radicals
In the following example we will solve a problem similar to Example 4 in general form.



Example 5. Multiply and simplify m n  x y m n  x y .
Solution. Let a  m n and b  x y . Then
m n  x y m n  x y   a  b a  b   a  b
= m n   x y   m  n   x  y   m n  x y .
2
2
Final answer:
m
2
2

2
2
2
2
2
2

n  x y m n  x y  m2n  x2 y .
Note. The expressions m n  x y and m n  x y are called conjugate to each other.
The above formula shows that the product of conjugate expressions is a rational number
(if, of course, all letters in these expressions represent rational numbers).
The above property of conjugate expressions can be used to simplify some fractions with
radicals in denominators (to rationalize the denominators). Namely, the following
fractions can be simplified
exp
m nx y
and
exp
,
m nx y
where exp is some expression. To rationalize the denominators, we multiply both
numerators and denominators of these fractions by expressions conjugate to
denominators.
Products and Quotients Involving Radicals
Example 6. Rationalize the denominator and simplify
4
.
5 3
Solution. We multiply numerator and denominator by expression
conjugate to the denominator 5  3 .



    
4 5  3  4 5  3 

 2 5  3   2 5  2
=
53
2
4

5 3



4 5 3
4 5 3

2
2
5 3 5 3
5  3
3.
5  3 , which is
Products and Quotients Involving Radicals
Example 7. Rationalize the denominator and simplify
x 3
.
3 x 2 y
Solution. Again, we multiply numerator and denominator by the expression 3 x  2 y ,
which is conjugate to the denominator 3 x  2 y .



x 3 3 x 2 y
x 3

.
3 x 2 y
3 x 2 y 3 x 2 y



Let’s process numerator and denominator separately.
Numerator:



x  3 3 x  2 y  x 3 x  x  2 y  33 x  3 2 y
= 3 x  x  2 x  y  9 x  6 y  3 x  2 xy  9 x  6 y .
Denominator:
3

   
2
x 2 y 3 x 2 y  3 x  2 y

2
 9x  4 y .
From here,
3 x  2 xy  9 x  6 y
x 3

.
9x  4 y
3 x 2 y
End of the Topic