lesson 1-9 properties of exponents...in pemdas, exponents occur before multiplication. 36 x2 4 * 9 *...
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Math-1010
Lesson 1-9
Properties of Exponents
Properties of Exponents
What is a power?
Power: An expression formed by repeated
multiplication of the base.
43xCoefficient Base
Exponent
The exponent applies to the number or variable
immediately to its left, not to the coefficient !!!
No Exponent? x3 113 xUsually, we don’t write the exponent ‘1’ (saves ink).
No Coefficient?3x 3*1 x 31 *1 x
Usually, we don’t write the coefficient ‘1’ (saves ink).
Negative?2x 2*)1( x 21 *)1( x
Usually, we don’t write the coefficient ‘-1’, we just
put the “negative symbol” (saves ink).
4x
Base Exponent
Factor: a number that is being multiplied.
means “base x used as a factor 4 times”
xxxxx ***4
Product: the result (“answer”) when multiplying.
Power: is repeated multiplication
xxxxx ***4
multiplication: is repeated addition
xxxx 3
How can you tell which definition to use?
Use the “context” of the problem
xx 43 (adding two terms)
)()( xxxxxxx
xxx 743
How can you tell which definition to use?
Use the “context” of the problem
22 32 xx (adding two terms)
)()( 22222 xxxxx
222 532 xxx
How can you tell which definition to use?
Use the “context” of the problem
32 * xx (multiplying two terms)
)**)(*( xxxxx
532 * xxx
Exponents
?3 2 x
?)3( 2 x
?)3(4 2 x
23x
)3)(3( xx
There’s no way to
simplify this anymore.
29x
)9(4 2x
In PEMDAS, exponents occur before
multiplication.
236x
2*9*4 x
Exponents
?2
2
x
?3
23
x
22
xx Remember how to
multiply fractions?4
2x
xxx 3
2
3
2
3
2327
8
x
Simplify2)4( y
2)5(2 x3
2
x
42
x216y
250x
4
16
x
8
3x
Product of Powers Property
))(( 32 xx
)**)(*())(( 32 xxxxxxx
5x
This is ‘x’ used as a factor how many times?
‘x’ used as a factor five times
3232 xxxWhen you multiply powers having the
same base, you add the exponents.
Power of a Power Property
)*)(*)(*()( 32 xxxxxxx
32 )(x6x
63*232)( xxx
32 )(x
This is ‘x’ used as a factor how many times?
‘x’ used as a factor six times
When you raise a power to another power
(“power of a power”)
you multiply the exponents.
Power of a Product Property 2)(xy
))(( xyxy yxyx *** yyxx ***
22 yx
This makes it seem like you can “distribute” in
the exponent. This only works with the power of a product!!
mmm yxxy )(
222)( yxyx
You must “FOIL” the power of a suml!!
))(()( 2 yxyxyx
22 2 yxyx
Combination of 1. Power of a Product2. Power of a Power
243 )3( yx
2431 )3( yx
8623 yx
‘x’ is a number, we just don’t know what it is. You treat all
numbers the same (whether they are variables or constants).
cmbmammcba yxyx 3)3(
Constants (integer, etc.) usually have an exponent of ‘1’.
512x
))((*4*3 32 xx
?)4(3 32 xx
You can re-arrange the order of
multiplication.
Coefficients of the powers are handled
separately from the base and the exponent.
Simplify:
)2)(5( 32 xx
3
2
1)2( xx
510x
4x
52 )2( x
43 2*)(5 xx
1032x
710x
35 )2( y158y
What is the difference between?
44 x)( andx
2332 )( and )( xx
4334 and xxxx
)1)(1( and )1( 2 xxx
Watch the negatives! 243 )( yx2431 ))1(( yx
862)1( yx This way you will be able to tell if the
simplified version is positive or negative. 86 yx
Turn negative signs into multiplication by -1.
362 )2( yx
3621)2( yx
18632 yx
Negative coefficients have an exponent of ‘1’.
1868 yx
A negative number raised to an odd
exponent remains negative.
Your Turn: simplify
534 )(2 xm
342 )2( zyx 31268 zyx
15202 xw
432 )2(3 yzx 1248x
Negative Exponent Property
2x
“Grab and drag”
1
*1 2
x
2
1
x
When you “Grab and drag” the base and its
exponent across the “boundary line” between
numerator and denominator, you just change the
sign of the exponent.
Why do we call this the negative exponent property?
22 yx2
2
y
xExamples:
2
3
1
x 6
1
x
6x6
1
x
Negative Exponent Property
24 x
Possible errors
1
*4 2
x
2
4
x
When you “Grab and drag” the base and its exponent across
the “boundary line” between numerator and denominator,
you just change the sign of the exponent.
DO NOT GRAB the coefficient!2
2
4
1
1
*4
x
x
Zero Exponent Property
120
1)2( 0 x
21*22 0 x
310 1000
210 100
110 10
010 1
Any base raised to the zero power simplifies to one.
Combination: (1) Negative Exponent, (2) Product of Powers, (3) Power of a Power, (4) Power of a Quotient
2
4
2
2
3
yx
x2
11
61
2
3
y
x2
42
2
3
y
xx2
6
2
3
y
x
2*12*1
2*62*1
2
3
y
x
22
122
2
3
y
x
2
12
4
9
y
x
Your Turn:52 )( w
2
32
2
3
2
zy
x
2332
1 x
1
2
4
3
5
y
x
10
1
w
62
9
x
4
64
4
9
x
zy
245
3
yx
Examples:
172
1032
yx
x
17
21032
y
xx
“Grab and drag”
17
21032
y
x
Product of powers: add
the exponents of same
based powers
17
832
y
x
Your Turn:
5
3
4
2
x
x
x
x
2
)2(9 4
3
42
2
)(
x
x
3
432
2
)2(
yx
xy
22
1
x
372x
52
1
x
9
78
x
y
yx
x4
2
2
3
Do you “grab and drag (up or down)??
yx
x4
2
2
3
y
xx
2
3 42
y
x
2
3 42
Product of powers property: add the
exponents of like-based powers
y
x
2
3 6
yxx 242
3
Product of powers property: add the
exponents of like-based powers
yx 242
3
yx 62
3
Make sure when you’re all done, there are
NO NEGATIVE EXPONENTS remaining.
y
x
2
3 6
It doesn’t matter!!!!
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