Interactive Algebra Review

2. Exponents and Radicals
Part B: Radicals and Rational Exponents

Note If you want to review integer exponents, go back to Part A: Integer 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 nonnegatve 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 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.

Here are some for you...

Here are some of the algebraic rules governing radicals.

Q1

12

is the same as

	4 3			2 + 8			3 2			2 3
	2 + 10

Q2

3   24

is the same as

	2   3   3			2 3			4   3   12			6
	-6

Q3

3     -1  8

is the same as

	1 2			- 1 2			- 1 24			-  2
	2

Q4

20 27

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, unless the denominator in the exponent is odd.)

Rational Exponents We can use fractional exponents for expressions involving radicals as follows: Radical Notation     Exponential Notation     Example  a a1/2   (or   a0.5) 641/2 = 8 3    a a1/3 641/3 = 4 n    a a1/n 321/5 = 2 In general, we can use the following rule: n   am = am/n           For example,   323/5 = (321/5)3 = 23 = 8   or n    a 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.

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) apaq = ap+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) (ap)q = apq (162)1/4 = 161/2   =   4 (d) (ab)p = apbp (4.2)1/2 = 41/221/2   =   22 (e) a b p     = ap bp 16 9 1/2     = 161/2 91/2 = 4 3

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))

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	=

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

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