RADICALS & RATIONAL EXPONENTS
Facts about Power Equations
Consider the power equation
xN = #,
with N > 1 integer and # any real number. Regarding the
solvability of
this equation, one has the following cases
I. If N is odd, the equation has one real solution, no matter what the
value of # is.
II. If N is even, then
(a) if # = 0 : the equation has one real solution x = 0;
(b) if # > 0 : the equation has two real solutions;
(c) if # < 0 : the equation has no real solutions.
EXAMPLE 1: The table below contains several equations, their solutions,
and the applicable cases
Equation |
Solution(s) |
Case |
x3 = 27 |
x = 3 |
I |
x5 = −32 |
x = −2 |
I |
x2 = 0 |
x = 0 |
II (a) |
x2 = 144 |
x = 12, −12 |
II (b) |
x4 = 625 |
x = 5, −5 |
II (b) |
x2 = −4 |
no real sol. |
II (c) |
Radical Notation
With N and # as above, the notation
![](./articles_imgs/4671/radica8.gif)
designates one special solution of the power equation xN =
#, namely:
I. If N is odd, the unique solution is chosen
II. If N is even, the non-negative solution is chosen.
This is applica-
ble only when # ≥ 0. In the case when # < 0, the radical is
undefined.
Convention: When N = 2, the radical is
simply denoted by ![](./articles_imgs/4671/radica58.gif)
Based on the discussion on the equations in Example
1, we have:
![](./articles_imgs/4671/radica9.jpg)
Properties of Radicals
Undoing Formulas for Radicals.
![](./articles_imgs/4671/radica10.jpg)
![](./articles_imgs/4671/radica11.jpg) |
if N is odd |
if N is even |
|
(UFR) |
Arithmetic Formulas for Radicals.
![](./articles_imgs/4671/radica12.jpg)
(Assuming N√a and N√b are defined.) |
(AFR) |
WARNING! What is incorrect about the equality
![](./articles_imgs/4671/radica13.jpg)
The above equality holds only when a ≥ 0. For instance, if
we try a = −12,
we have
![](./articles_imgs/4671/radica14.jpg)
By (UFR2) we know that ![](./articles_imgs/4671/radica15.jpg)
TIP: The above formulas can be used when we want to simplify
![](./articles_imgs/4671/radica16.jpg)
where Expression involves powers, products and quotients.
We do so by
• factoring N-powers out of Expression,
• then pulling N-powers outside using (UFR)
and (AFR).
EXAMPLE 2: Suppose we want to simplify
![](./articles_imgs/4671/radica17.jpg)
Using the equalities
and we see that we
can factor the square under the radical, then
we pull it out:
![](./articles_imgs/4671/radica21.jpg)
NOTE: Since N = 2 is even, we pulled out the square using
the absolute
value. If all variables x, y, z are assumed to be positive, the above simplifi-
cation can be continued (with last equality optional) as:
![](./articles_imgs/4671/radica22.jpg)
Fractions with radicals
Suppose we have a fraction, such as the one in Examples 3 and 4 below,
and we are asked to transform it (by multiplying both sides by the same
Expression) into an equivalent one
![](./articles_imgs/4671/radica23.jpg)
so that the new fraction is ”nicer.” The specification of
what ”nicer” means
is often formulated as a request:
• rationalize the denominator, or
• rationalize the numerator.
This means that we look for and expression, such that one of the products
(Numerator) ·(Expression) or (Denominator)· (Expression) is without radicals.
EXAMPLE 3: If we want to rationalize the denominator in
the expres-
sion is pretty obvious: (using ):
![](./articles_imgs/4671/radica30.jpg)
TIPS: For more complicated Numerators or Denominators, try
one of these
identities (based on (a + b)(a − b) = a2 − b2):
Rationalizing tricks.
![](./articles_imgs/4671/radica36.jpg) |
(RT) |
EXAMPLE 4: To rationalize the denominator in
![](./articles_imgs/4671/radica31.jpg)
we multiply both sides by
. Note that in the new numerator we
have a square, for which we employ the Square of a Sum Formula ((a+b)2 =
a2 + 2ab + b2):
![](./articles_imgs/4671/radica33.jpg)
Fractional Powers (Rational Exponents)
Suppose is a rational number (with the
fraction simplified). The ![](./articles_imgs/4671/radica35.gif)
powers are computed using one of the equalities:
provided
is defined
In particular, for an integer N > 1, the
powers are computed by
provided
is defined
EXAMPLE 5: To compute we use the definition with M = −3 and
N = 5, combined with (UFR2) and the equality 32 = 25:
![](./articles_imgs/4671/radica37.jpg)
EXAMPLE 6: To convert
to radical notation, we use the second equal-
ity in the definition (for we use M = 3, N =
4; for we use M = −1,
N = 4), then we simplify (last equality optional) using (AFR1):
![](./articles_imgs/4671/radica41.jpg)
Arithemtic of Fractional Powers
FACT: The formulas first introduced in R2, collected in the table below, also
work if the epxonents m and m are rational.
Arithmetic Formulas for Powers.
![](./articles_imgs/4671/radica42.jpg)
(Provided all powers are defined.) |
(AFP) |
EXAMPLE 7: To simplify
we add the exponents using (AFP1):
![](./articles_imgs/4671/radica44.jpg)
(How did we get When
adding and
we converted them to fractions
with denominator 6, that is: )
EXAMPLE 8: To convert
to rational exponents, we replace directly
the operation with the
power, then (note that 16 = 24) we use (AFP):
![](./articles_imgs/4671/radica52.jpg)
EXAMPLE 9: To convert to
radical notation, we first simplfy (see
also R2 for the technique of handling products and quotients of powers) the
expression using (AFP), then convert the factors to radicals :
![](./articles_imgs/4671/radica54.jpg)
(Make sure you understand the calculations of
and ![](./articles_imgs/4671/radica56.jpg)
|