Interavtive Algebra Review

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(This topic can be found in Section 0.2 of the books Applied Calculus and Finite Mathematics).

0.2 Exponents and Radicals

Previous Tutorial: Part A: Integer Exponents
This tutorial: Part B: Radicals and rational exponents

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 42 = 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 24 = 16. We can similarly define sixth roots, eighth root, and so on.

Q What about odd-numbered roots?
A There is a slight difference with odd-numbered 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 23 = 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 odd-numbered roots are defined in the same way.

Notation We use "radical" notation to designate roots, as shown below.

Radicals

Name Notation       Example
Square root of a          
a

16
 = 4

1
= 1

-1
  is not a real number

2
  = ...   An irrational number
Cube root of a
3
a
3
8
 = 2
3
-27
 = -3
3
1
 = 1
3
-1
 = -1
Fourth root of a
4
a
4
16
 = 2
4
81
 = 3
4
-1
  is not a real number

Here are some for you...

Here are some of the algebraic rules governing radicals.

Radicals of Products and Quotients

In the following identities, a and b stand for any real numbers. In the case of even-numbered roots, they must be nonnegative.

    Rule Example
    n
    a b
    =
    n
    a
    n
    b

    8
    =

    (4)(2)
    =

    4

    2
    =

    2

    2
    n
    a

    b
    =
    n
    a


    n
    b
    3
     8

    27
    =
    3
    8


    3
    27
    =
    2

    3

 

Q1 is the same as

Q2
is the same as
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
a1/2   (or   a0.5)
641/2 = 8
a1/3
641/3 = 4
a1/n
321/5 = 2

In general, we can use the following rule:

= am/n           For example,     323/5 = (321/5)3 = 23 = 8
 
or
m
 
 

 

Q What entitles us to use fractional exponents for radicals?
A If we want to make any sense of, say, 91/2, 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
RuleExample
(a)
a^p a^q = a^{p+q}
85/38-1/3  =   84/3   =   (81/3)4   = 24  =   16
(b)
ap

aq
= ap-q   if a ≠ 0    
93/2

92
= 9-1/2 =   1/91/2  =  1/3
(c)
(a^p)^q = a^{pq}
(162)1/4 = 161/2   =   4
(d)
(ab)^p = a^pb^p
(4.2)1/2 = 41/221/2   =   2>2
(e)
a

b
p
 
 
=
ap

bp
16

9
1/2
 
 
=
161/2

91/2
=
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:
axm/n

bxp/q
or \frac{a\sqrt[n]{x^m}}{b\sqrt[q]{x^p}}
where all exponents are non-negative.

Examples: Expressions in Rational Form:
2.1x2

2x1/4
,  
4

7x3/4
,   \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 axn where n is any exponent (possibly negative, zero, or not an integer).

Examples: Expressions in Exponential Form:
4.1x-3/2 ,  
2

3
x1/2 ,  
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:
Rational FormExponential Form
\sqrt{x} x^{1/2}
\sqrt[3]{x^2} x^{2/3}
\frac{2\sqrt{x}}{3} \frac{2}{3}x^{1/2}
\frac{1}{\sqrt{x}} x^{-1/2}
\frac{2}{3.5\sqrt{x}} \frac{2}{3.5}x^{-1/2}
\frac{2.1}{\sqrt[3]{x^2}} 2.1x^{-2/3}
\frac{4}{5\sqrt[3]{x}} \frac{4}{5}x^{-1/3}
\frac{4}{5\sqrt[3]{x^2}} \frac{4}{5}x^{-2/3}
2.3\sqrt{x}
 
\frac{2.3}{\sqrt{x}}
 
\frac{2\sqrt[5]{x^2}}{3}
 
3.2x-1/2
 
-4

3
x-3/2
 

Simplify each of the following so that the result contains no negative exponents.

Q1
x1/3

x4/3
=    
Q2
(x2)5/3

x1/3
=    
Q3
x-1/2y-3/2

x3/2y3/2
=    
Q4
2y-1/2

x1/2y-5/2
-2
 
 
=    

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