1. THE THINK TANK · SPRING 2016 · DIFFICULT PROBLEM FOR MATH 100
IVAN RODRIGUEZ
ABSTRACT. The aim of this article is to explain how to simplify a particular expression
involving a cube root. This exercise is interesting because this solution invokes the differ-
ence of cubes formula.
In this exercise, we are asked to simplify
41/2
+ 31/3
41/2 − 31/3
.(1)
To begin, we first recall the difference of cubes formula:
x3
− y3
= (x − y) x2
+ xy + y2
.(2)
Here, we notice that the denominator of (1) resembles the (x − y) piece in (2); thus, we
set
x = 41/2
and y = 31/3
.
Next, we multiply both the numerator and denominator of (1) by the x2
+ xy + y2
piece:
41/2
+ 31/3
41/2 − 31/3
=
41/2
+ 31/3
x2
+ xy + y2
41/2 − 31/3 (x2 + xy + y2)
=
41/2
+ 31/3
41/2 2
+ 41/2
31/3
+ 31/3 2
41/2 − 31/3 41/2 2
+ 41/2 31/3 + 31/3 2
.(3)
At this point, we observe that the denominator of (3) is equal to x3
− y3
according to
(2). Thus, we clean up the denominator of (3) as follows:
41/2
− 31/3
41/2
2
+ 41/2
31/3
+ 31/3
2
= x3
− y3
= 41/2
3
− 31/3
3
= 43/2
− 33/3
= 8 − 3
= 5.
Date: 24 April 2016 and, in revised form, 24 April 2016.
Key words and phrases. Mathematics, MATH 100, preparation for university-level math, simplify, simplify-
ing, simplification, expression, cube root, difference of cubes, conjugate, hard, difficult, challenging.
1

2. 2 IVAN RODRIGUEZ
From here, we now clean up the numerator of (3):
41/2
+ 31/3
41/2
2
+ 41/2
31/3
+ 31/3
2
=
41/2
+ 31/3
42/2
+ 41/2
· 31/3
+ 32/3
=
2 +
3
√
3 4 + 2
3
√
3 +
3
√
9 =
8 + 4
3
√
3 + 2
3
√
9 + 4
3
√
3 + 2
3
√
3
3
√
3 +
3
√
3
3
√
9 =
8 + 4
3
√
3 + 2
3
√
9 + 4
3
√
3 + 2
3
√
9 +
3
√
27 =
8 + 3 + 4
3
√
3 + 4
3
√
3 + 2
3
√
9 + 2
3
√
9 =
11 + 8
3
√
3 + 4
3
√
9.
Putting the pieces together, the numerator of (3) is 11+8 3
√
3+4 3
√
9 while the denominator
is 5. Hence,
41/2
+ 31/3
41/2 2
+ 41/2
31/3
+ 31/3 2
41/2 − 31/3 41/2 2
+ 41/2 31/3 + 31/3 2
=
11 + 8 3
√
3 + 4 3
√
9
5
.
Consequently, we have ‘simplified’ (1):
41/2
+ 31/3
41/2 − 31/3
=
11 + 8 3
√
3 + 4 3
√
9
5
.
DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF ARIZONA, TUCSON, ARIZONA 85721
E-mail address: ivanrodriguez@email.arizona.edu