properties of positive integer exponents, m > 0, n > 0 for example,
TRANSCRIPT
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Properties of Positive Integer Exponents, m > 0, n > 0
• for example,
• for example,
• for example,
,nmnm aaa
,)( mnnm aa
,)( mmm baab
.222 532
.2)2( 623
.32)32( 222
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More Properties of Positive Integer Exponents, m > 0, n > 0
• for example,
• for example,
• for example,
,m
mm
b
a
b
a
,, nmaa
a nmn
m
.2
6
2
62
22
.22
2 32
5
,,1
nmaa
anmm
n
.2
1
2
235
2
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Properties of Integer Exponents, a 0
• for example,
• for example,
• for example,
,10 a .12
10
,1
mm
aa .
2
12
33
,1m
m
aa .
2
12
33
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Scientific Notation• A number is written in scientific notation if it is of the form
where 1 a < 10, and m is some integer.
• Scientific notation is useful for writing very large or very small numbers since fewer zeros are required in the representation of the number.
• Example. One light-year is about 6 trillion miles, which is 6,000,000,000,000 miles using decimal notation. This may be written as using scientific notation.
,10ma
12106
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More about Scientific Notation• If the result of a measurement or calculation is written in
scientific notation as where 1 a < 10, and m is some integer, then the number of digits of a are taken as the significant digits of the result.
• If we write a number in scientific notation with fewer significant digits than the original number presented, we must round the last significant digit according to the rule: add 1 to last significant digit if digit following it in the original number is 5, 6, 7, 8, or 9
leave the last significant digit alone if digit following it in the original number is 0, 1, 2, 3, or 4
• Example. The speed of light is 186,282 miles per second. This is 1.863 105 to four significant digits.
,10ma
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Summary of Integer Exponents; We discussed
• Six properties for positive integer exponents
• Three properties for general integer exponents
• Scientific notation
• Significant digits
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Rational Exponents and Radicals• If n is a natural number and n is odd, b1/n can be defined to be
the unique nth root of b.
• If n is a natural number and n is even, b1/n can be defined to be the unique positive nth root of b. In this case, we say that b1/n is the principal nth root of b.
• Now, we define bm/n for an integer m, a natural number n, and a real number b, by
where b must be positive when n is even. With this definition, all the rules of exponents continue to hold when the exponents are rational numbers.
nmmnnm bbb /1/1/ )()(
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Examples for Rational Exponents
• Simplify 274/3. We have
• Simplify We have
• Simplify . We have
• Simplify 163/4. We have Note that 161/4 is taken to be the principal root, which is 2.
.813)27(27 443/13/4
.)(4
41222/1
a
bbaba .)( 222/1 ba
12
6/5
3/23/1
z
yx12
6/5
3/23/1
z
yx .10
84
z
yx
.82)16(16 334/14/3
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Radicals
• The principal nth root of a real number b was discussed previously. An alternative representation of this root is available using a radical symbol, . We summarize as follows.
with these restrictions:
if n is even and b < 0, is not a real number, if n is even and b 0, is the nonnegative number a satisfying an = b.
• Warning.
baabb nnn where,/1
n b n b
.24 ,24
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Radicals and Exponents
• We have that
• Example.
• Problem. Change from rational exponent form to radical form, Assume that x and y are positive real numbers.
Solution.
.)()(
and ,)(/1/
/1/
mnmnnm
n mnmnm
bbb
bbb
???)8()8(
8)8(82323/1
3 23/123/2
.)( 2/5yx
.)()(552/5 yxyxyx
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Properties of Radicals
• for example,
• for example,
• for example,
,m
nn m bb .882
33 2
,nnn abba .3694
,nn
n
b
a
b
a .
27
8
27
83
3
3
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More about Radicals
• For example,
• For example,
• Simplify
. odd, is If aan n n .2)2(3 3
.|| even, is If aan n n .2 |2| )2( 2
.55555
3335
44444
444
.5
3
5
3
5
3
)5(5
)3(3
55555
3335 5
5 5
55
5
54
4
544444
444
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Rationalizing Denominators
• A fraction is sometimes considered simplified if its denominator is free of radicals. The process by which this is accomplished is called rationalizing the denominator.
• In this connection, a useful formula is:
• Example. Rationalize the denominator:
.nmnmnm
.25
4
253
4
25
254
25
25
25
4
25
4
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Examples for Simplifying Expressions Involving Radicals
• .
•
.
8/72/14/74/32/1 xxxxxxxxxx
x
x
x
xx
xx
xxxxxxxxx
x
22
12/12/12/12/12/12/1
2
)1(1212
21
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Rational Expressions and Radicals; We discussed
• nth roots and the principal nth root
• rational exponents
• radicals
• converting from rational exponent form to radical form and vice versa
• properties of radicals
• rationalizing the denominator