B. R~gkgl§Jor roots) ancUractional exponents. 1. Definition. \ the positive nth root
of a if n is even and a) 0 n.fa denotes the nth root of a if n is odd. When a is ...
Algebra: notation,
I.
exponents, simplifying
scientific expressions
---------------------------------------------------------------------
For more practice problems and detailed written explanations, lowing books, both on reserve all year in the Sciences Library.
see the fol-
by Loren C. Larson (can be purchased in the bookstore) .Al.g,e.braancl1rigonometry, by Keedy and Bittinger
--------------------------------------------------------------------A. Exponents:
1. Definition
Definitions and rules.
al = a,
a2 = a-a,
2. aman = am+n m
a
n
(provided
o
4. .I&f a = 1 (provided
5. Drl a
-n
a
(reason: a
$
1 =~ a
_=
a C) 3 [ a
a-a-a-a-a a-a-a
~ = 1 and a
a =a a
)
= a 5-3 = a 2
1-1
1
0 =a )
0)
(a
$
o
1
(reason: - n = L n = a
0)
a
O-n-n =a
)
a
An example showing why:
6. {ab)n = an bn (
8. Note: (a + b)n
$
= a2 + 3 = as
= (a-a)(a-a-a)
a2a3
5
=a m-n
an = a-a..- a (n times)
An example showing why: (
3 . -a
a3 = a-a-a,
;II!:
(ab)3 = ababab = aaabbb = a3b3 ]
an + bn !!! (except when n = 1 or a = 0 or b = 0) 1.1
I
e.g. (1+2)3= Correct
33= 27,while
rules:
13+23=1+8=9.
= a2
(a + b)2
+
2ab + b2 ,
(a + b)3 = a3 + 3a2b + 3ab2 + b3 , etc.; see Section D .
9. Exponentiation precedes multiplication'. For example, 7a3 = 7.a.a.a , nQ1 (7a)3 , which would be 7a.7a.7a 10. (-a)n
= (-l)nan
=
an if n even -an
{
= (-x)(-x)(-x) = (-)(_)(_)x3 = -x3
(-x)3
Note: -x2
=
(-)x.x
5
- -3 x
=
~
(-x)2 .
1 5-1 =__1. 1 --
Examp~
x3
if n odd
= (-x)(-x) = (_)(_)x2 = x2
e.g. (-x)2
1
= 73a3 .
.'
(x
5
-2 3
10
-3
= 1
1000
I
) =x
-6
= 0.001.
8 -4 4 xx =L= 2 x x
;
'
x
2
' '
3x4. 5x3 = (3.5)(x4.x3) = 15x7 5 -1 or x y
Exercises I A Simplify each expression. 1. 2x3. 3x2 2. -(2x3)4 3. (-4x-1z-2)-2 4. (5x4y-3z2)(-2x2y4z-1) -2 3 -3 x 6 . -1y 2 5. 4 +-12-4 8 x Y -2 -4 3 7. (3a -lb -2c4)3 8 . 5 -3-5 x y (2a -lb2 c-3)2 2 x y
1.2
;
B. R~gkgl§Jor
roots) ancUractional
1. Definition
exponents
the positive nth root of a if n is even and a) 0
n.fa
denotes
\
the nth root of a if n is odd
When a is negative and n is even (e.g. 2./-9 ), n.;a is undefined within the real number system. ~: .;a is short for 2.fa, the positive square root of a. Examples: 2ff = ff = 3; 3ff = 2; 3./-8 = -2; ./-9 is undefined.
2. Definition
a1/n
is defined to be n.;a
(where possible)
.
(1. n) 1/n n n 1 1/n . (a ) =a = a = a , so a ~ the nth root of a)
(Reason:
am/n = (a1/n)m
3. Definition
value). provided
4. n.;ab = n.;a. nJD
.QI: (am)1/n
(these have
the same
a 1/n is defined.
and
nJa71) = n.;a/ nJD
provided n.ra- and nJD are both defined. (Note
./(-2)(-8)
5. n.;a+b e.g. ./32
= ./I6 = 4 even though j:2 and ;:a
n.;a + nJD.
~
+
42
In particular,
./a2
+ b2
~
are undefined.) a
+ b ,
= ./9 +16 = /25 = 5 , while 3 + 4 = 7 .
6. jX2 = IxI , the absolute value of x. Examp~ 1. 84/3 = (3jS)4 = 24 2. 182/3 = 3'/18"2"" = 3'/(3.372)(3.3.2)
3.
R
14
2
1L = :i..x:..= 1L 9
19
3
but
= 16 = 3.3'/2.3.2" = 3.3/12
J?
12
1L = :i..£ = ill 9
19
4. J62 = 6; ./(-6)2 = 6; j12 = j4:3 = /4n-=
3
2n-
5. Simplify 3fi2TaP-. 1.3
I
3j72a'1)9- (72a7b9)1/3 . (72)1/3a7/3b9/3 :I 81/391/3a7/3b3 = 2.91/3a7/3b3 or 2a2b3.3./9a (either of these is OK) 6. Rationalize the denominator of
l
= 5.13 = sJ3
13 13.13
or
3
13
(eliminate the radical)
.
~ 3
7. Simplify (0.027)1/3. (0.027)1/3
=(
27 )1/3
1000
=
= ( 333 )1/3
10
(33)~/3 (103 )1/3
= i-
or
0.3
10
Exercises La Simplify: 1. 163/2 2. (0.008)1/3 3. [(-3)(2)]1/4 5. x-3/2 / x3/2 4. (x2/3)(xl/3)4 6.
J-25 (~-64
,.
7.
~ x3'
9.
v 3/4
2/3
1/2
(x Y exponents:
Express the following using rational 8.
( x 1/3
JJ;
)2
)
10.
x~
1
Rationalize the denominator:
20
12
11.
J5
12.
c. Scientific notation Scientific notation is a uniform way of writing numbers in which each number is written in the form k times 10n with 1:s k
