

Radicals
(roots) and/or exponentiation
with fractional (rational) exponent

Rules and properties of radicals

Rules and properties of radicals
and/or fractional exponents

Simplifying radical expressions

Operations
on radical expressions 
Adding, subtracting and multiplying
radicals

Rationalizing a denominator

Radicals
and/or fractional
(rational) exponents 





Radicals
(roots) and/or exponentiation
with fractional (rational) exponent


The
n^{th}
root of a real number a,
denoted 

is defined
to be such number x
that x^{n}
=
a, 

or 

where
n
is a natural number greater than 1. 

Thus, if a
= 0,
then 


if
a
> 0,
then 

is the positive real number
x such
that x^{n}
=
a, 

if
a
< 0
and n
is odd, then 

is the negative real number
x such that
x^{n}
=
a, 

if
a
< 0
and n
is even,
then 

is not a real number,


if n
is an even number, the n^{th}
root of a
is defined to be that
number x
≥ 0 such that x^{n}
=
a.

Even roots exist only
for nonnegative numbers.

Odd
roots always exist and have the property that 

where n
is odd.


The expression 

is called a
radical, the number a
is called the radicand, and n
is the index of the radical.


The symbol 

is called the radical
sign.


The principal n^{th}
root of a real number a
is the unique real number x
which is an n^{th}
root of a
and is of the same sign as a.


Rules and properties of radicals

Let n
be a natural number
and a
be a real number, then

1. 


2. 


3. 



Examples: 
1. The odd root of any real number exists: 

a) 

since
3^{3}
= 27. 
b) 

since
(3)^{3}
= (3)·(3)·(3)
= 
27. 
c) 

since
(2)^{5}
= (2)·(2)·(2)·(2)·(2)
= 32. 

2.
The
even root of a nonnegative real number exists:


a) 

since 4^{2}
= 16 and 4
> 0, [recall
that ( ±
4)^{2}
= 16]. 
b) 

since 3^{4}
= 81 and 3
> 0, [recall
that ( ±
3)^{4}
= 81]. 

Note that the even root is defined to be a nonnegative
number.


3.
The
even root of a negative real number does not exist as a real
number. 

For example, 

do
not exist as real numbers, but they do exist as complex
numbers. 


Rules and properties of radicals
and/or fractional exponents

If
m,
n and p
are natural numbers
(n
>1) and
if a
and b
are nonnegative
real numbers, then 
Properties 

Examples 


























Simplifying radical expressions

As,
to simplify means to reduce given expression to a simpler form, or find another expression with the same
value. So, for example a square root radical expression is simplified when its radicand has no square factors.

To simplify a radical means to
remove factors from the radical until no factor in the radicand
has an exponent greater than or equal to the index of the radical
and the index is as low as possible.

Therefore, use the product rule of
radicals to simplify the
n^{th} root radicals.
Throughout,
we take variables to be positive, unless the index of the
radical is odd.



Operations
on radical expressions

Throughout,
we take variables to be positive, unless the index of the
radical is odd.

Adding, subtracting and multiplying
radicals

Examples: 













Rationalizing a denominator

Rationalizing a denominator is a method for changing an irrational denominator into a rational one. 
Examples: 








since a^{3} 
b^{3}
=
(a

b)
· (a^{2}
+ ab
+ b^{2}) 


Radicals
and/or fractional
(rational) exponents 











Intermediate
algebra contents 



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