## Interactive Algebra Review

(Based on the algebra reviews in Calculus Applied to the Real World, , and Finite Mathematics and Calculus Applied to the Real World and )

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## 2. 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 positive number whose fourth power is a. Thus, the fourth root of 16 is 2, since 24 = 16. We can similarly define sixth roots, eigth root, and so on.

Answer There is a slight difference with odd-numbered roots: The cube root of a real number a is the 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 wame way.

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

Name Notation Example
Square root of a
 a
 16     =   4
 1     =   1
Cube root of a
 3a
 38  =  2
 3-27   =   -3
Fourth root of a
 4a
 416  =  2
 481   =   3
Here are some for you...
 3125  =

 5-1  =

 9 + 16  =

Here are some of the algebraic rules governing radicals.

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

 Rule Example

Rather than doing any more work with radical expressions, we will first convert all radical notation to exponential notation, as follows. (Throughout, we take a to be positive, unless the denominator in the exponent is odd.)

Rational Exponents

We can use fractional exponents for expressions involving radicals as follows:

 a
 a1/2   (or   a0.5)
 641/2 = 8
 3a
 a1/3
 641/3 = 4
 na
 a1/n
 321/5 = 2
In general, we can use the following rule:
 nam or na m
 am/n
 323/5 = (321/5)3 = 23 = 8

Question What entitles us to use fractional exponents for radicals?
Answer 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 saure root of 9. A fuller explanation is given in the texts Calculus Applied to the Real World, and Finite Mathematics and Calculus Applied to the Real World.

Question Do all the usual rules for exponents work with frational exponents?
Answer Yes. Here is a summary of these rules (the same as those we saw in the previous topic).

Exponent Identities
RuleQuick Example
(a)
 apap = ap+q
85/34-1/3  =   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)
 (ap)q = apq
 (162)1/4 = 161/2   =   4
(d)
 (ab)p = apbp
 (4.2)1/2 = 41/221/2   =   22
(e)
 ab p = apbp
 169 1/2 = 161/291/2 = 43

Simplify each of the following so that the result contains no negative exponents, fill in the blanks, and press "Check." (Where necessary, use formula format, for example x^2/y^4,   x^2*y^4 or 1/(x^2*y^4))

 (x2)5/3x1/3 =
 x-1/2y-3/2x3/2y3/2 =
 2y-1/2x1/2y-5/2 -2 =

Now go over the examples and try some of the exercises in Section A.2 of the Algebra Review of Calculus Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Alternatively, press "next" button on the sidebar to go on to the next topic.

Last Updated: December, 1998