Root, cube root and nth root
Let’s say if you multiply the same number twice you get 9. What is the
original number? The answer can be 3 or −3, since
3 × 3 = (−3) × (−3) = 9
(1)
In such a case, we express:
3=
√
9
(2)
√
We pronounce 9 as “root 9” or “square root 9.” Of course −3 can be
the answer to our previous question, but as a root cannot take multiple
values as the result, we take only non-negative one as the answer. (Imagine,
if this were not the case. Then, if you try to calculate root 9 by pressing
buttons in a calculator, you will get 3 and −3 as answers. This doesn’t make
much sense, as a calculator cannot spit out two answers on the screen. This
concept of taking a single value instead of multiple values will be clear when
we talk about the concept of function later.)
In other words, if x is non-negative number that satisfies x2 = y. We
have
√
x= y
(3)
(Problem 1. Evaluate the followings:)
√
0,
√
9,
√
100,
√
r
49,
1
4
(4)
In most cases, you have
√ to resort to calculator to calculate a root. For
example, if you calculate 2 you will find:
√
2 = 1.414213562 · · ·
(5)
On the other hand, the examples in Problem 1 were carefully chosen so that
you don’t need to use the calculator to find the answer.
The root satisfies some properties. First,
√ √
√
x y = xy
(6)
One can show this as follows:
√ √
√
√
( x y)2 = ( x)2 ( y)2 = xy
1
(7)
Similarly, one can show
√
r
x
x
√ =
y
y
One also has:
√
(8)
z 2 = |z|
(9)
where |z| denotes the absolute value of z. This formula is obvious when z is
positive or zero. In such cases |z| is given by z. It is also not that hard to
check
the formula
when z is negative. For example, when z = −5 we have
√
p
2
2
(−5) = 5 = 5 = | − 5|.
Using these relations, we can play around with roots. For example,
√
√ √
√
√
√ √
√
8 = 4 2 = 2 2,
18 = 9 2 = 3 2
(10)
√
Now, let me introduce cube root. If x3 = y, then x = 3 y. For example,
√
√
3
as 23 = 8, we have 2 =
√ 8. Notice that this is not equal to 3 8, which
means 3 multiplied by 8. The 3 in cube root is written small.
There is also another big difference between the root and the cube root.
Notice that the root of a negative number doesn’t exist; if you multiply the
same number twice you get always non-negative number. (Remember if you
multiply −3 by −3 you get 9 not −9.) On
√ the other hand, the cube root of
a negative number exists. For example, 3 −27 = −3 as (−3)3 = −27.
We can actually generalize the square root and the cube root to nth root.
√
For example, if xn = y is satisfied, we have x = n y.
Finally, let us mention how the concept of root, cube root and nth root
connects to exponent. First, notice that expressions such as x1/2 wouldn’t
make much sense at first glance, since you cannot multiply a number “half”
times. However, there is a way to assign a value in a consistent way. (Remember, we assign values to cases in which a number is multiplied “0” times
and “negative” times. There is nothing we can’t do.) So, let’s see. Observe:
1
1
(x 2 )2 = x 2 ·2 = x1 = x
Therefore, we conclude
1
x2 =
√
x
(11)
(12)
Similarly, one can show:
1
x3 =
√
3
1
x,
xn =
√
n
x
(13)
Given this, what would expressions like x2/3 mean? We have:
More generally,
√
2
1
x 3 = (x 3 )2 = ( 3 x)2
(14)
√
p
√
x q = ( q x)p = q xp
(15)
2
Problem 2. Simplify or evaluate the following. (Hint1 )
4
3
8 =?,
√
√
12 3 =?,
√
18 −
√
8 =?,
4
− 32
=?,
− 1
2
1
=?
4
Problem 3. Simplify followings:
√
ab
b
1
!2
=?,
a b
× =?,
b c
√
√ √ √
√ √
18 = 9 2, 8 = 4 2.
3
a3/2
=?
ab
(16)