Interavtive Algebra Review
(This topic can be found in Section 0.2 of the books Applied Calculus and Finite Mathematics).
0.2 Exponents and Radicals
If a is any nonnegative real number, then its square root is the nonnegative number whose square is a. For example, the square root of 16 is 4, since 4^{2} = 16. Similarly, the fourth root of the nonnegative number a is the nonnegative number whose fourth power is a. Thus, the fourth root of 16 is 2, since 2^{4} = 16. We can similarly define sixth roots, eighth root, and so on.
Q What about oddnumbered roots?
A There is a slight difference with oddnumbered roots: The cube root of a real number a is the unique number whose cube is a, so that, for example, the cube root of 8 is 2 (since 2^{3} = 8). Note that we can take the cube root of any number, positive, negative or zero. For instance, the cube root of 8 is 2, since (2)^{3} = 8. Unlike square roots, the cube root of a number may be negative. In fact, the cube root of a always has the same sign as a. The other oddnumbered roots are defined in the same way.
Notation We use "radical" notation to designate roots, as shown below.
Here are some for you...
Here are some of the algebraic rules governing radicals.
Q3 
3 
 1
8 

is the same as 
Q4 
 20
27 

is the same as 
Exponential Notation
Rather than working all the time with radical expressions, we can convert all radical notation to exponential notation, as follows. (Throughout, we take a to be positive if the denominator in the exponent is even.)
Rational Exponents
We can use rational exponents for expressions involving radicals as follows:
Radical Notation 

Exponential Notation 

Example 












In general, we can use the following rule:

= a^{m/n}
For example, 32^{3/5} = (32^{1/5})^{3} = 2^{3} = 8

or 


^{m} 


Q What entitles us to use fractional exponents for radicals?
A If we want to make any sense of, say, 9^{1/2}, and have the laws of exponents continue to work, we are forced to define it as the square root of 9.
A fuller explanation is given in the texts Applied Calculus and Finite Mathematics and Applied Calculus.
Q Do all the usual rules for exponents work with fractional exponents?
A Yes. Here is a summary of these rules  the same as those we saw in the previous topic  but this time we permit the exponents p and q to be rational numbers (rather than integers as in the last tutorial).
Exponent Identities
Rule  Example 
(a) 

8^{5/3}8^{1/3} = 8^{4/3 } = (8^{1/3})^{4} = 2^{4} = 16 
(b) 
a^{p}
a^{q} 
= 
a^{pq} if a ≠ 0 

9^{3/2}
9^{2} 
= 
9^{1/2} = 1/9^{1/2} = 1/3 

(c) 

(16^{2})^{1/4} 
= 
16^{1/2} = 4 

(d) 

(4^{.}2)^{1/2} 
= 
4^{1/2}2^{1/2} = 2>2 

(e) 


16
9 

^{1/2} 
= 
16^{1/2}
9^{1/2} 
= 
4
3 


Rational and Exponential Form with Radicals
Extending the ideas of the previous tutorial, we will say that an expression of rational form is an expression written as a ratio: 
ax^{m/n}
bx^{p/q} 
or 
\frac{a\sqrt[n]{x^m}}{b\sqrt[q]{x^p}} 

where all exponents are nonnegative.

Examples: Expressions in Rational Form: 
2.1x^{2}
2x^{1/4} 
, 
4
7x^{3/4} 
, 
\frac{1}{2\sqrt{x}} 
, 
\frac{2\sqrt[3]{x}}{3x} 
, 
\frac{3x^2}{\sqrt{x}} 

but 
\frac{3x^2}{\sqrt{x^{2}}} 
is not in rational form because the exponent of x in the denominator is negative. 

An expression of exponential form is an expression written as ax^{n} where n is any exponent (possibly negative, zero, or not an integer). 
Examples: Expressions in Exponential Form: 
4.1x^{3/2} 
, 
2
3  x^{1/2} 
, 
2
5.1  x^{4/3} 
, 
22
7 

but 
4
7x^{1/2} 
is not in exponential form because of the x is in the denominator. 

Converting between Rational and Exponential Form Using Exponent Identities:

Simplify each of the following so that the result contains no negative exponents.
You should now try the examples and exercises in Section 0.2 of the Algebra Review of Applied Calculus and Finite Mathematics and Applied Calculus
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Last Updated: january, 2008
Copyright © 2007 Stefan Waner