Download Exponents and Scientific Notation

College Algebra - MAT 161
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Copyright 2009 Killoran
Exponents and Scientific Notation
32 D 9 but this mistake is
The one major mistake with exponents is the thought that
caused by not seeing the order of operations involved
32 D
D
D
1 32
1 9
9
This is another reason that I like replacing variables with .
/
To help us better understand the workings of exponent expressions a little notation and
English interpretation is needed:
34 D 3 3 3 3 in English this would be "four factors of 3"
4n D 4 4 4
4 or "n-factors of 4"
When we have a m a n , we can interpret this to be:
"m-factors of a" times "n-factors of a"
1
1.1
giving a total of "m+n-factors of a"
Rules of Exponents
Product Rule:
a m a n D a mCn
1.2
Quotient Rule:
am
D am
an
1.3
zero exponent:
an
D an
an
1.4
n
n
D a 0 also
an
D 1 so a 0 D 1
an
Negative exponent:
a3
D a3
a7
7
Da
Thus: a
n
D
4
also
a3
1
D 4 so a
a7
a
1
1
and
D an
an
a n
4
D
1
a4
College Algebra - MAT 161
1.5
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Copyright 2009 Killoran
Power Rules:
.a m /n D a mn
a
b
m
D
am
bm
.a m bn / p D a mp bnp
1.6
a
b
Reciprocal Rule
1
D
b
a
There is no "order" to these rules, any can be used at any time in the simplification of an
expression, but it is best to simplify inside of . / whenever possible
Example 1 Simplify: 3x 2 y 3
2x 3 y 4
3x 2 y 3
2x 3 y 4 D .3 2/ x 2 x 3
D 6x 5 y 7
Example 2 Simplify: 2x y 2
3
2x y 2
3
D 23 x 3 y 2 3
D 8x 3 y 6
Example 3 Simplify
15x 3 y 5
20x 6 y 2
15x 3 y 5
15 x 3 y 5
D
20 x 6 y 2
20x 6 y 2
3 1 y3
D
4 x3 1
3y 3
D 3
4x
Example 4 Simplify: 2x 2
2
3x y 2
2x 2
2
3x y 2 D 4x 4
D 12x 5 y 2
3x y 2
y3 y4
College Algebra - MAT 161
Example 5 Simplify: 3x 4 y
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Copyright 2009 Killoran
2
3
3x 4 y
2
3
D3
2
x
2.4/
D3
2
x
8 6
D
2. 3/
y
y
y6
32 x 8
y6
D 8
9x
Example 6 Simplify:
2
3a 2 b 3
12a 5 b2
2
3a 2 b 3
12a 5 b2
D
12a 5 b2
3a 2 b 3
D
4a 5 b2 b3
a2
D 4a 3 b5
2
2
2
D 16a 6 b10
Example 7 Simplify:
13m 23 n 7
26m 3 n 10
0
13m 23 n 7
26m 3 n 10
0
D1
anything raise to the zero power equals one!
Example 8 Simplify: 2x
5
27C3x
2x
5
27C3x D 2.x
5/C.7C3x/
D 24xC2
Example 9 Simplify:
53xC4
53 2x
53xC4
D 5.3xC4/
53 2x
D 55xC1
Example 10 Simplify: 43x
.3 2x/
1 2x
43x
1
2x
D 4.3x
D 46x
D 22
D 212x
1/ 2
2
6x 2
4
College Algebra - MAT 161
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Copyright 2009 Killoran
Simple Scientific Notation:
3:2 1023 D 3:2E 23 on most scientific calculators. It is NOT an Error message. It is
just telling you that the number could not be represented in 8-10 digits (too many zeros for
place holders)
3:2 1023 D 320; 000; 000; 000; 000; 000; 000; 000 a very LARGE number (positive
exponent)
1:6
10
8
D 0:000000016 a very SMALL number (negative exponent)
To work with calculator there is either an EE button. Just type in the first part 3:2 choose
the EE button and then enter the exponent .23/. Be careful to use the negative . / button
for negative numbers not the minus button :