
Skills to Master
·
Use and interpret rational exponents with 1 in the numerator
·
Use and interpret
rational exponents whose numerators are not 1
·
Use and interpret
radical expressions and convert between radical expressions and rational
exponents
·
Simplify and combine
radical expressions
·
Name a polynomial and
identify its degree
·
Add, subtract, and
multiply polynomial expressions
·
Rational Exponents Whose Numerators Are 1
Suppose
you were asked to determine the value of the numerical expression
. Not knowing the meaning of rational (or
fractional) exponents, you might try applying the power-to-power rule:
. Hence you conclude that if
and
is a natural number, then
is the non-negative real number
such that
. That is,
is the nth root of
. For the above example, if
,
then
. That is,
is the number that, when multiplied by itself,
produces 25. Clearly,
. We see, then, that fractional exponents
produce the roots of numbers.
Examples
since
. The square root of 16 is 4
since
. The cube root of 8 is 2
Remember
that the square root of a number produces two values: one positive and one
negative. But with a rational exponent,
we use the principal root, which is
always positive.
since
. The 4th root of 81 is 3 and the 4th
root of
is x. But since both x and –x
produce the same answer, we use take the absolute value to guarantee that the
result is positive.
Also,
if
is even and
is negative, then there is no real number
answer. However, if
is odd and
is negative, then the answer is the negative nth root of b.
Examples
has no real number solution since, if
is positive, there is no real number such that
(Why?)
since ![]()
Rational Exponents Whose Numerators Are Not 1
Remember
that fractions are multiplied by multiplying the numerators together and the
denominators together. That is,
. We
can use this to factor fractions as well.
For example,
.
Keeping this in mind and using the power-to-power rule for exponents, it
is easy to see that, for example,
.
Or we can write
.
So for a rational expression
, as an exponent it is either the nth root raised to the power of m, or as the power of m raised to the nth root. (Note that this is
the result of the commutative property of multiplication. Why?)
In general, for real number b and whole numbers m and n,
.
We can use the same line or reasoning to interpret
:
OR ![]()
Example

Radical
Expressions
Another way to express the root of a number is to
use the radical sign. The radical sign is
. In general,
and
. Here, b is called the radicand and n is the index of the expression. For
n = 2, the index is understood. That is,
.
Examples
since ![]()
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An important fact to keep in mind is that
. That is, taking the nth root of a number raised to the nth power, or raising an nth root to the nth power cancels the root and returns the radicand (or the
base). Notice that this makes sense
since
. We will use this fact
when solving equations that contain powers or roots.
Examples
;
; ![]()
Simplifying
and Combining Radicals
For real numbers a, b and n:
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This
means that the multiplication of two nth
roots can be written as the nth root
of the product of the radicands, and the division of two nth roots can be written as the nth
root of the quotient of the radicands.
Note that this only works if the values of n are the same!!!
Example
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Since
the roots of most numbers are irrational, we often leave a radical expression
in radical form. So rather than 1.414…,
we write
. This allows us to combine the same radicals as
a multiplication. For example,
(i.e., three times the square root of seven). In general, for real numbers a, b,
c and n,
. Note that this only works when the radicals
have the same value. It is not possible,
for instance, to simplify
since n
and m are not equal (assuming that
they are not equal). Nor can we simplify
since a
and b are not the same (assuming they
are not equal). However, if a or b
can be factored such that one of the factors is a perfect power of n and the other is the same for both a and b, then we can simplify the expression. For example, ![]()
Examples
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Polynomials
A
monomial is a number or the product of a number and one or more
variables with whole-number exponents.
The number is called the coefficient
of the monomial.
Examples
Coefficient
=
Coefficient =
Coefficient =
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The
degree of a monomial is the sum of
the exponents of its variables. All
non-zero constants have a degree of zero.
For the examples above, what is the degree of each monomial?
A
monomial or a sum of monomials is called a polynomial. Each monomial in a polynomial is called a term of the polynomial.
Examples
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Monomials |
Binomials |
Trinomials |
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The
degree of a polynomial is the degree of the term with highest degree.
Any
terms in a polynomial that have the same variables with the same exponents are
called like terms. The important thing to remember about like
terms is that they can be added or
subtracted.
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Like Terms |
Unlike Terms |
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Like
terms are combined by adding or
subtracting their coefficients.
Examples
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Multiplying Monomials
Examples
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Multiplying a Monomial and a Polynomial
Use the distributive property.
Examples
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Multiplying Two Binomials
Use the FOIL method: First Last Inside Outside
Examples
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