Division of Monomials and Scientific Notation
A. Division of monomials or Integer Exponents
Example: Simplify
x3y _ ^_y_ _^£y_ orx3-i y_ _ w
x2z2
x2 z2
x-tf z2
z2
z2
\
8x2
8 GZZ
2 1
2
——X'tf'fi'fi'A?*)-. x2
&££!
x7
— = X^^-= X2
Xs
Rule for dividing exponential expressions
Ifm and 7i are two integers and x * 0, then
Example: Simplify
r s
r°s
__
Y
5.
Y
5
>t
r
2-1
5
=
7-3
4Z3
6,
^
3 2
4
, 3 h
v
s
x5 _ 1//>f£^
W?f* x-x-x
XB
x
x-x-x
.
-3
X'
*
-3
-/= —
X
X
Negative exponents
If n is a positive integer and x ± 0, then
v—71 _ J_
xn
—
e.,4. '
v77
and/—- = X
-•b
i
X3^x »
—
Example: Simplify
v2
2--$"""
-"*,
y
—---
I
_
.3
-^
*
3
3
XI
JL
_x_
i
5"
21 - X-
"$-1
X
3a:-1y4
3±£
X
6-lx3y-2
x
.»*?. i£ =lit
j<i
t
(-3x4y~5)(-2x~3y_1)
^y
(l
= (o X
6x
0
a
*5
rftw
*
x5-5=x0
X° = 1
= I
5°^ L
Xs
Zero as an exponent
If X * 0, then x° = 1. The expression 0° is UNDEFINED.
Example: Simplify
o
3& ^i
T(36)°=rl
C~2)-o
(x"V)°=
3.0
x
o
-*&) =
o
3 - x •* •
1-111
Exercise 1:
a) Evaluate 5 3.
—
) .
I
f3
b) Evaluate -^3>
9
c) Simplify:
4
a
d) Simplify:
3
a
vr
a
VJ
9
V J
w
o
3 2
1m
v y
-7
5
//;? ry
2
6
—lo-rx,
e) Simplify: (-7.v3)(^5)2. ~7y^ „. y '__ JJ^r\
J7ir$—1
•V
r ,-2
-5
f) Simplify: -A^.
16c »
5
i^c'u:^
(uj
B. Scientific Notation
a) Write the number 4^00^000^000,000,in scientific notation.
^e &X (0
4,600,000,000,000 = 4.6 x 10^
b) Write the number 0 000043 in scientific notation.
0.000043 =4.$x 10"5
/_ -
. -5"
" ^ X 10
nv^T"-^ ^u.^c^^u-€ -^/c/jCmt?^
Very large and very small numbers are encountered in the fields of science and
engineering.
In scientific notation, a number is expressed as a product of two factors, one a number
between 1 and 10 and the other a power of 10 To change a number written in decimal notation
to scientific notation, write it in the form a x 10 /where a is a number between 1 and 10 and n is
an integer.
Example:
Write the number 0.000054 in scientific notation.
"5
Write the number in decimal notation. 2.8 X 103
Write the number 5.1x10 2 in decimal notation.
5j xio
- 0. obi
0.000000 ...OOab = atb X 10~n negative exponent
a^00000...q0>= a.b x 10" positive exponent
-•V
Q#^X io
I
- o. oooo.,. oao
<^
w
QbXl0 ~ ^ °°° •r'°^°
Exercise 2:
i)
Write the number 4.7xlO12 in decimal notation.
^100,000,000,000
J
ii)
Write the mass of Mars, which is approximately 6^2,000,000,000,000,000,000,000,
kg, in scientific notation.
6^2X i0
iii)
The diameter of a dust particle is 0,QQQQQQQg2 meters. Write this number in scientific
notation.
iv)
a
Write the mass of Earth, which is approximately 5^80,000,000,000,000,000,000,000
kg, in scientific notation.
v)
^
^,„
.,,
-—-"
The electric charge on an electron is 0.000000000Q00000Q001£ coulomb. Write this
number in scientific notation.
v—
Exercise 3:
_
/
/ -£V *)
1. Multiply: (3,8 x10(9,7 xia8) =-\b-8)(^l) fc[l<0j/[«0,/
= h&2(bX\o
02. Divide:
_. .. 2.356xl09—
= -b.&SG x /o* ov
? hi Co y -J—l°^
= 5=' J°^S- >(
6.2x10"*
Q>^
3. Multiply: (3.0 x 105)(1.1 x 10-8)
^ Divide:
r^- -j 7.2X1013
4.
2.4X10"3
—
-
-? V
> A
,^. '£
•
\0
Jf
%&<o
AM (*x -ax-f) -h(V""&*+ 0
">^- iT? -J-
MuX^ (Vx^ £)(2 *frV'J(7 x~|.
- 2 -2
'<f
70 x
*,<>
9
•
3x u £
-12 Xw3^
3 ,3
-'^
• T
3
,\7"
1-3
*
M-l
A
£0. b o')
A a
-i
3
2r
eA3/2>£?
4 a2" 6
-12 ^-2)
7
zfe
-10
/o
a.
lo
_- £
1
lo
IX -2-
4a a
(X
10