
ALGEBRA
 solved problems







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

Rules and properties of radicals


67. 
Find given roots or radicals.


Solutions: 
a) 

since
3^{3}
= 27. 

b) 

since
(3)^{3}
= (3)·(3)·(3)
= 
27. 

c) 

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

The odd root of any real number
exists. 


Solutions: 
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]. 

The
even root of a nonnegative real number exists.
Note that the even root is defined to be a nonnegative
number.

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


69. 
Simplify radical expressions.




Operations
on radical expressions 
Adding, subtracting and multiplying
radicals


70. 
Solve given radical expressions.


Solutions: 





71. 
Solve given radical expressions.




72. 
Solve given radical expressions.




73. 
Solve given radical expressions.




74. 
Solve given radical expressions.












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