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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.

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...

 25
is a real number and equal to
is not a real number.

 3 -125
is a real number and equal to
is not a real number.

 -25
is a real number and equal to
is not a real number.

 7 -1
is a real number and equal to
is not a real number.

 1/4
is a real number and equal to
is not a real number.

 9 + 16
is a real number and equal to
is not a real number.

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 ab
=  n a n b
 3 827
=  3 8 3 27
=
2

3

 Q1 is the same as

 4
 2 +
 3
 2
 +
Q2
is the same as
 2
 2
 4
 6
 -6
Q3
 3 -18
is the same as
 12
 - 12
 12
 -
 -
Q4
 2027
is the same as
 2 10 9 3
 10 2 9 3
 2 10 3 3
 4 10 9 3
 2 5 3 3

## 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.

 Q1 43/2 = Q2 1252/3 = Q3 (1/8)5/3 = Q4 (-1)5/7 =

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)
 apaq = ap-q   if a ≠ 0
 93/292 = 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)
 ab p = apbp
 169 1/2 = 161/291/2 = 43

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

Examples: Expressions in Rational Form:
 2.1x22x1/4 , 47x3/4 , \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 , 23 x1/2 , 25.1 x-4/3 , 227
 but 47x-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
 -43 x-3/2

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

Q1
 x1/3x4/3
=
Q2
 (x2)5/3x1/3
=
Q3
 x-1/2y-3/2x3/2y3/2
=
Q4
 2y-1/2x1/2y-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