Radicals and Rational
Exponents
Lecture #2
H.Melikian/1100/04
Dr .Hayk Melikyan
Departmen of Mathematics and CS
melikyan@nccu.edu
1
Definition of the Principal Square Root
In general, if b2 = a, then b is a square root of a.
If a is a nonnegative real number, the
nonnegative number b such that b2 = a, denoted by
b =  a, is the principal square root of a.

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Square Roots of Perfect Squares
For any real number a
2
a a
In words, the principal square root of a2 is the
absolute value of a.
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The Product Rule for Square Roots

If a and b represent nonnegative real number,
then
ab  a b and

a b  ab
The square root of a product is the product of
the square roots.
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Text Example

Simplify a. 500
b. 6x3x
Solution:
a.
500  100 5
b.
6x  3x  6x  3x
 100 5
 18x 2  9x 2 2
 10 5
 9x 2 2  9 x 2 2
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 3x 2
5
The Quotient Rule for Square Roots

If a and b represent nonnegative real numbers
and b does not equal 0, then
a
a

b
b

and
a

b
a
.
b
The square root of the quotient is the quotient
of the square roots.
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Text Example

Simplify:
100
9
Solution:
100
100 10


9
9
3
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Example

Perform the indicated operation:
43 + 3 - 23.
Solution:
4 3 32 3 3 3
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Example

Perform the indicated operation:
24 + 26.
Solution:
24  2 6 
2 62 6 4 6
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 Rationalizing
the denominator:
If the denominator contains the square root
of a natural number that is not a perfect square,
multiply the numerator and denominator by
the
smallest number that produces the square root
of a perfect square in the denominator.
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What is a conjugate?
Pairs of expressions that involve the sum & the
difference of two terms
 The conjugate of a+b is a-b
 Why are we interested in conjugates?
 When working with terms that involve square
roots, the radicals are eliminated when
multiplying conjugates

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Definition of the Principal nth Root of a Real Number
n

a b
n
means that b  a
If n, the index, is even, then a is nonnegative
(a > 0) and b is also nonnegative (b > 0) . If n
is odd, a and b can be any real numbers.
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Finding the nth Roots of Perfect nth Powers
If n is odd, a  a
n
n
If n is even a  a.
n
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n
13
The Product and Quotient Rules for nth Roots

For all real numbers, where the indicated roots
represent real numbers,
n
a b  ab and
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n
n
n
n
a n a

, b0
b
b
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Definition of Rational Exponents
a1 / n  n a .
Furthermore,
1
1
1/ n
a
 1/ n  n , a  0
a
a
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Example

Simplify 4 1/2
Solution:
1
2
4  4 2
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Definition of Rational Exponents
a

m/ n
m
n
m
( a)  a .
n
The exponent m/n consists of two parts: the
denominator n is the root and the numerator m is
the exponent. Furthermore,
a
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m/n

1
a
m/n
.
17

If z is positive integer, which of the following is
equal to 2 16 z
2
a. 32 z
2
b. 12z
2
c. 8 z
d. 8z
e. 4z
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POLYNOMIALS:
The Degree of axn.
If a does not equal 0, the degree of axn is n.
 The degree of a nonzero constant is 0.
 The constant 0 has no defined degree.

A polynomial in x is an algebraic expression of the
form
anxn + an-1xn-1 + an-2xn-2 + … + a1n + a0
where an, an-1, an-2, …, a1 and a0 are real numbers.
an != 0, and n is a non-negative integer.
The polynomial is of degree n, an is the leading
coefficient, and a0 is the constant term.
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Text Example
Perform the indicated operations and simplify:
(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)
Solution
(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)
= (-9x3 + 13x3) + (7x2 + 2x2) + (-5x – 8x) + (3 – 6) Group like terms.
= 4x3 + 9x2 +(– 13)x + (-3)
Combine like terms.
= 4x3 + 9x2 - 13x – 3
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Multiplying Polynomials
The product of two monomials is obtained by using properties of
exponents. For example,
(-8x6)(5x3) = -8·5x6+3 = -40x9
Multiply coefficients and add exponents.
Furthermore, we can use the distributive property to multiply a
monomial and a polynomial that is not a monomial. For
example,
3x4(2x3 – 7x + 3) = 3x4 · 2x3 – 3x4 · 7x + 3x4 · 3 = 6x7 – 21x5 +
4.
9x
monomial
trinomial
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Multiplying Polynomials when Neither is a Monomial

Multiply each term of one polynomial by each
term of the other polynomial. Then combine like
terms.
Using the FOIL Method to Multiply Binomials
last
first
(ax + b)(cx + d) = ax · cx + ax · d + b · cx +
inner
outer
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Product of
First
terms
b·d
Product of Product of Product of
Outside Inside terms Last terms
terms
22
Text Example
Multiply: (3x + 4)(5x – 3).
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Text Example
Multiply: (3x + 4)(5x – 3).
Solution
last
first
F
O
I
L
(3x + 4)(5x – 3) = 3x·5x + 3x(-3) + 4(5x) + 4(-3)
= 15x2 – 9x + 20x – 12
inner
= 15x2 + 11x – 12Combine like terms.
outer
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The Product of the Sum and Difference of Two Terms
2
(A  B)(A  B)  A  B

2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
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The Square of a Binomial Sum
2
2
2
(A  B)  A  2AB  B

The square of a binomial sum is first term
squared plus 2 times the product of the terms
plus last term squared.
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The Square of a Binomial Difference
2
2
2
(A  B)  A  2AB  B

The square of a binomial difference is first
term squared minus 2 times the product of
the terms plus last term squared.
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Let A and B represent real numbers, variables, or algebraic expressions.
Special Product
Sum and Difference of Two Terms
Example
(A + B)(A – B) = A2 – B2
(2x + 3)(2x – 3) = (2x) 2 – 32
= 4x2 – 9
Squaring a Binomial
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
(y + 5) 2 = y2 + 2·y·5 + 52
= y2 + 10y + 25
(3x – 4) 2 = (3x)2 – 2·3x·4 + 42
= 9x2 – 24x + 16
Cubing a Binomial
(A + B)3 = A3 + 3A2B + 3AB2 + B3
(A – B)3 = A3 – 3A2B + 3AB2 - B3
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(x + 4)3 = x3 + 3·x2·4 + 3·x·42 + 43
= x3 + 12x2 + 48x + 64
(x – 2)3 = x3 – 3·x2·2 + 3·x·22 - 23
= x3 – 6x2 – 12x + 8
28
Example
x2 – y2 = (x - y)(x + y)
x2 + 2xy + y2 = (x + y)2
x2 - 2xy + y2 = (x - y)2




A. if x2 – y2 = 24 and x + y = 6, then
x–y=


B. if x – y = 5 and x2 + y2 = 13, then

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-2xy =
29
Text Example
Multiply: a. (x + 4y)(3x – 5y)
b. (5x + 3y) 2
Solution
We will perform the multiplication in part (a) using the FOIL method. We will
multiply in part (b) using the formula for the square of a binomial, (A + B) 2.
a. (x + 4y)(3x – 5y)
Multiply these binomials using the FOIL method.
= (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y)
= 3x2 – 5xy + 12xy – 20y2
= 3x2 + 7xy – 20y2
Combine like terms.
F
• (5 x + 3y) 2
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O
I
L
= (5 x) 2 + 2(5 x)(3y) + (3y) 2
= 25x2 + 30xy + 9y2
(A + B) 2 = A2 + 2AB + B2
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Example

Multiply: (3x + 4)2.
Solution:
( 3x + 4 )2
=(3x)2 + (2)(3x) (4) + 42
=9x2 + 24x + 16
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