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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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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)

With N and # as above, the notation

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

Based on the discussion on the equations in Example
1, we have:

 if N is odd if N is even
(UFR)

(Assuming N√a and N√b are defined.)

(AFR)

WARNING! What is incorrect about the equality

The above equality holds only when a ≥ 0. For instance, if we try a = −12,
we have

By (UFR2) we know that
TIP: The above formulas can be used when we want to simplify

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

Using the equalities   and we see that we
can factor the square under the radical, then we pull it out:

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:

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

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 ):

TIPS: For more complicated Numerators or Denominators, try one of these
identities (based on (a + b)(a − b) = a2 − b2):

 Rationalizing tricks. (RT)

EXAMPLE 4: To rationalize the denominator in

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):

Fractional Powers (Rational Exponents)
Suppose is a rational number (with the fraction simplified). The
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:

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):

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. (Provided all powers are defined.) (AFP)

EXAMPLE 7: To simplify we add the exponents using (AFP1):

(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):

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 :

(Make sure you understand the calculations of and