Download Write as a power. (0.06) (0.06) (0.06) (0.06) Step 1: Identify the term

4X 2y
TlOWT
Write as a power.
Write as a power.
(0.06) (0.06) (0.06) (0.06)
(1.7) (1.7) (1. 7)
Step 1: Identify the term that is multiplied repeatedly.
The term (0.06) is repeated several times in the
expression. This term is being multiplied by itself.
It is the base.
Step 2: Count the number of times the repeated term is in
the expression.
(0.06) (0.06) (0.06) (0.06)
1
2
3
4
The repeated term is included 4 times.
The exponent is 4.
Step 3: Write the exponential expression.
Step 1:
Identify the term that is multiplied repeatedly.
---
is the base.
Step 2: Count the number of times the repeated term is in
the expression.
_ _ _ is the exponent.
Step 3: Write the exponential expression.
Put the base within parentheses. Put the exponent
outside.
(0.06)4
Answer: (0.06) (0.06) (0.06) (0.06)
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= (0.06)4
Answer: (1.7)(1.7)(1.7) = _ __
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4x2y
[ Simplify.
I Simplify~ -4 2 -
-52 + (-3)3 - 14
Step 1: Simplify the exponents.
Remember that a negative sign stays with a number
if the negative sign is included within parentheses.
The exponent tells you the number of times to
multiply the number.
- 52 = - (5 . 5) = - 25
(_ 3)3 = (-3· -3· -3) = -27
14 = 1 . 1 . 1 . 1 = 1
-4 2
+
(~3)3
© 2002 , Renaissance Learning, Inc.
- 14
=
(_2)4 = - - - - -
(-1)3 =
+
(_2)4
-4 2
(_1)3
------- + ------­
Simplify from left to right.
-25 + -27 - 1
-52
- 1 = -53
Answer: _52
+ (-1)3
Step 1: Simplify the exponents.
-52 + (-3)3 - 14 -25 + -27 - 1 Step 2:
(_2)4
=
-53
Step 2: Simplify from left to right.
Answer: -4 2
-
(_2)4
+
(-1) 3 =
_________
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4x2y
Simplify.
("65)
I
- 3
Simplify.
Step 1: Review negative exponents.
(
~
r
Step 1:
Review negative exponents. To drop the negative sign on the exponent, how should the number being raised to a power change? Step 2:
negative power.
Take the reciprocal of the number being raised to a negative power. (~r3 = (~r
(~) -2=
Negative exponents tell you to take the reciprocal of
the number being raised to a power.
(ifl = (if (~) - 2
=
Gf Step 2: Take the reciprocal of the number being raised to a
Step 3: Write the product without the exponent and multiply.
(~r
Answer:
=
("65)-3
=
~.~ .~
216
125
© 2002 , Renais sance Learning, Inc.
=
__
Step 3:
Write the product without the exponent and multiply.
Answer:
("29)
i~~
- 2
=
_ _ _ __
California •
4x2y . Tnrn
ISimplify.
I~imp\ify
( - 8)1
Step 1: Review fractional exponents.
The denominator of a fractional exponent tells you
what root to take. Two in the denominator tells you
to take the square root of the number. Three in the
denominator tells you to take the cube, or third, root
of the number.
The numerator of a fractional exponent works the
same as a whole number exponent.
3
8
3 b -3. = -yo
3!.8bB
a?
- =!-ya
Step 2: Rewrite the expression as a radical.
;';-8
-8
Write the following as radical expressions.
5
/4 =
Step 2:
7
-----
25 2
=
-----
Rewrite the expression as a radical.
2
= _ _ __
=
-2
= (_ 2)2
Step 4: Raise the number to a power.
( - 2) 2
same as __________________
Step 3: Take the root of the number.
Step 3: Take the root of the number.
;/(_ 8)2
The numerator of a fractional exponent works the
(100)
(-8) 3 ~ ;/(_8)2
=
Step 1: Review fractional exponents.
The denominator of a fractional exponent tells you
3
2~
-2· -2· -2
(lOo)l
= -
2.- 2
=
Step 4: Raise the number to a power.
4
3
2
Answer: (-8)3 = 4
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Answer: (100)
2
= _ _ _ __
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4x2y
I
Simplify.
Step 1:
(
55. 23 . 48)5
56. 25 . 43
I
-­
Simplify.
Review properties of exponents.
When exponential expressions have the same base, the exponents can be simplified.
When dividing exponential terms, exponents on the
same base can be subtracted.
55 = 5(5 - 2) = 53
~
52 = 5(2-5)
~
= 5-3 =
Step 1:
~
~
43
_
-
(8 - 3)
4
=4
_
-
the exponents.
_ _ _ _ _ _ _ _ _ _ _ _ the exponents.
Step 2:
55.23. 48 _ 55 . 23 . 48
56'25'43 - 56 25 43
55 _ 5(5 - 6) _ 5- 1
6 5
(3 - 5)
23
-2
=2
25 = 2
Review properties of exponents. When dividing exponential terms that have the same
When a power is raised to a power,
l
Simplify the fraction.
1
4
base,
When an exponential term is raised to a power, multiply the two exponents. (53)5 = 5(3'5) = 515 Step 2:
74. 3 9 • 22
( 710 • 38 • 25)
Simplify the fraction. 74 '3 9 '2 2 1
710 . 38 • 25 ...L Simplify each individual fraction.
5
= 22 5
55'23.48 _
45
56 . 25. 43 - 5· 22 Step 3:
Raise the fraction to a power.
55'23'48)5 =
( 56. 25 . 43
Answer:
3 8
(55. 2 . 4 )5
56'25'43
© 2002. Renaissance Learning. 'Inc.
(---L)5=
5 . 22
25 = _54
5 '2
10
5
(4 ),5
(5 . 22)5
Step 3:
Raise the fraction to a power. 425
55~1O 2)4 _
74. 39'2
( 710 '38 '25
Answer:
-
----
=
(774.'339
.'222)4 _
10
8
5
­
California
4x2y I TII1111 Rewrite x- s using only positive exponents.
Step 1: Review negative exponents.
Negative exponents tell you to take the reciprocal of a
number. When a variable has a negative exponent, you
can change it to positive exponent by changing where
it is in a fraction.
3
5
.!2(£
_5_
a- 3
1
Rewrite ~ using only positive exponents.
x
Step 1: Review negative exponents.
When you switch a term with a negative exponent
from the numerator to the denominator of a fraction,
_______ the negative sign on the exponent.
= 5a 3
5a- 3 = ~3
a
When you move a term with a negative exponent
to the opposite part of the fraction, drop the
negative sign.
Step 2: Rewrite the expression using only positive exponents.
x-S -- x-1
s
© 2002, Renaissance Learning, Inc. Step 2: Rewrite the expression using only positive exponents.
1_
x- 8
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4x2y Ffnrn
I Multiply.
l
Multiply. (5x 4y) (~7/)
(3x 3yS) (7x:/)
Step 1: Simplify exponents with the same base.
Step 1:
The two x-terms can be combined. They are
exponential expressions with the same base.
The exponents can be added. The two y-terms
can be combined, too.
x3 • X
=
=
l·y2=
Simplify exponents with the same base.
X4 •
x7 = x(-
y . y3 =
(x· x . x) . x X(:3 + 1)
=
y<-
+ --)
+ -)
-
=
x
Y
X4 (y.y.y.y.y)(y.y)
= y(S + 2) = y7
Step 2: Multiply the whole numbers and write the
new monomial.
3·7 = 21
(3x 3yS) (7x:/)
=
21x 4y 7
Step 2:
Multiply the whole numbers and write the
new monomial.
5·4= _ __
4
(5x y)(4x
© 2002, Renaissance Learning, Inc.
y)
7
= --------­
California
4x2Y Objective-Multiply algebraic: expressions with fractional oxponent
1
~
x 3 (x 6
Multiply.
Step 1:
].
1
+ x4)
Use the distributive property.
Multiply the term outside the parentheses by the terms inside the parentheses.
213
21
X 3 (X 6 + X 4 ) .~ X 3 (X 6 )
Step 2:
-
23
-
7
-1
Use the distributive property. Multiply the term outside the parentheses by the
terms inside the parentheses.
1
+ X 3 (X 4 )
7
X 5 (x 10
Combine exponential terms having the same base.
2
Step 1:
-
x 5 (X 10 + X 2 )
Multiply.
1
+ x 2) = _______________ Step 2:
Combine exponential terms having the same base.
Answer:
XS
1
The x terms in x 3 (x 6) can be combined because the exponents have the same base, x. Combine the terms
by adding the fractions.
~
X3
1
(x 6 )
(2 61)
~ + 1
3
3
5
(2 1)
1
+
4
12
5
}
Answer: x 3 (x 6 + X 4)
© 2002 . Renaissance Learning. Inc.
4
8
6
2
+
9
17
12
12 3)
(
X3"+4
X 3" + 6 = X 6
~
4) = x 3"
~ + J.
6
±
+ 1
6
6
(2 :1)
~].
X 3 (x
=X 3" +
~
= X6 +
17 =X12 11 X 12 l.l
!
(x 10 + x 2 )
=
---------------------------
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4x2y
ETIT'!
Simplify. - 24ily2 - 8X 5y 8 Step 1: Simplify exponents with the same bases.
The two x-terms can be combined. Since the terms
are being divided, the exponents can be subtracted.
The two y-terms can be combined, too.
x3
x5
=
Step 1: Simplify exponents with the same bases.
-xx 5
8
,x . .¥.,X ,x ..¥.;X. x· x =
x
y2 _
ys
-30X 5y 7 5 x 8y 3 Simplify.
(3 - 5)
=
x
-2
-
-
/
X( - - ­
(---)
::-:r=y
Y' 1
--)
=
_ _ _ __
= x2
~ '-1 y . y. Y ',5'-5' y' y. Y =y(2 -
8)
=y - 6
1
y
= (;
Step 2: Divide the whole numbers and write the new monomial.
-24 -
-8 ­
_~y8
3
~6
­
x-y
© 2002, Renaissance Learning, Inc.
Divide the whole numbers and write the new monomiaL
-30
-5­
3
-24ily2 _
Step 2:
-30xY
5X 8 y 3
California
4x2y
-IS 4 - 12
Simplify.
42x y z -:-6x 4y 7z6 Simplify.
Step 1: Make all exponents positive.
To change a negative exponent to a positive
exponent, switch it to the other part of the fraction.
Make sure to move its base, too.
For example,
3x -- 3
3ys
-=-5
-3 .
Y
x'
In this monomial, move the x- and z-terms in the
numerator to the denominator. Move the x-term
in the denominator to the numerator.
42x - ISy 4z -6x
4/z 6
-
Step 3:
denominator? - - - - - - Which terms in the denominator should be moved to
- 4 -5- 6
3x Y z
-6XIS/z6z12
Step 2: Simplify exponents that have the same base and
are being multiplied.
42x4l -
-6x IS/z I8 Simplify exponents that have the same base and
are being divided.
Subtract the exponents: x:s = X(4 - IS) = x- ll = --h
x
L7
Y
42x4y4
-6x 15y 7z 18
Step 1: Make all exponents positive.
Which terms in the numerator should be moved to the
18x7y - 8z - 9
Step 2: Simplify exponents that have the same base and
are being multiplied.
Add the exponents: Z6Z 12 = Z(6 + 12) = Zl8
42x4l
-6xlSy7izl2
I
the numerator? _ _ _ _ _ __
42x4l
12
~
18x7y - 8z - 9
3x -4y 5z6
Step 3: Simplify exponents that have the same base and
are being divided.
x
=y (4 -7) =y-3
1
=-3
Y
42
-6x ll iz l8
Step 4: Divide the whole numbers and write the new monomial.
Step 4: Divide the whole numbers and write the new monomial.
~
-6 ­-~7
lz-6x- 4/i
42x- 15
12
© 2002, Renaissance Learning, Inc.
-7 xllizl8 California
4x2y
[ Simplify.
I Simplify.
(- 5b 4c- 8d 3) - 2
Step 1: Raise each term to the power outside the parentheses.
(_ 5b 4c- 8d 3 ) -2 = (_ 5) - 2 . (b4) - 2 . (C- II) - 2 . (d 3) -2
Step 2: Simplify powers raised to powers.
Remember that you can simplify powers raised to
powers by multiplying.
(-5) -2. (b 4) - 2 . (C - 8r-2 . (d 3) - 2
(-5) - 2. b- 8
• C 16 •
d- 6
Step 1:
(3b - 6c 2d 5
r-
3
Raise each term to the power outside the parentheses.
(3b - 6C 2d 5 ) - 3 = - - -
Step 2: Simplify powers raised to powers.
(-5) - 2b - 8C I6d- 6
=
Step 3: Make all exponents positive.
Put negative exponents in the denominator
of a fraction.
Step 3: Make all exponents positive.
16
C ·
(-5) -2b- Il CI6 d- 6 = (-5)2
8 6
bd
Step 4: Simplify any remaining numerical terms.
Step 4: Simplify any remaining numerical terms.
(-5)2 = 25
16
C
16 C
(-5)2 b8d 6
8 6
25b d
Answer: (-5b 4c- 8d 3)-2
= ~ll
25b d
© 2U02, Renaissance Learning, Inc. 6
Answer: (3b - 6C2d 5 ) - 3 =
California
4x2y
.
•
SImplify.
C
I Simplify.
(8b 3c10d - 5 ) - 4 (2b6c -4d 3) - 5 Step 1: Raise the numerator and the denominator to the
Step 1: Raise the numerator and the denominator to the
power outside the parentheses.
power outside the parentheses
(8b3c10d ~ 5)-4
8-4b-12c - 40d20
=
(2b6c - 4d3) -5
(3b 7c- 3d- 2 ) - 3
2 - 5b - 30C20d - 15
Step 2: Make all exponents positive.
(5b -- 3C5d-2) - 4
Switch bases with negative exponents to the opposite
part of the fraction.
8-4b-12C-40d20
25b30d20d15
2 - 5b - 30C20d - 15 -
84b12C20/0
r
(3b 7c- 3 2 ) - 3
(5b - 3c5d 2) 4
=
------------------------
Step 2: Make all exponents positive.
Step 3: Simplify exponents that have the same base and are
being multiplied.
Add the exponents:
= d(20+ 15) = d 35
c c = C(20 + 40) = COO
2 15 d 0d
Step 3: Simplify exponents that have the same base and
20 40
are being multiplied.
25b30d35
4 12 60
.2 5b30d20d15
84b12C2DC40
8b
C
Step 4: Simplify exponents that have the same base and are
being divided.
b30
(30 Subtract the exponents: b12 = b
25b30d35
4 12 60
12)
Step 4: Simplify exponents that have the same base and
are being divided.
18
= b
25b18d35 4 60 8c
8 b c
Step 5: Simplify any remaining numerical terms.
25
32
1
Step 5: Simplify any remaining numerical terms.
84 = 4096 = 128 25b18d35
4 60
8c
b18d 35
60
128c
bUli~5
(8b3c10d - 5) - 4
Answer:
(2b 6 c -- 4d 3) -5
© 2002, Renaissance Learning. Inc.
=
128c60
Answer:
r
(3b 7c- 3 2 ) - 3
(5b -3 c5d - 2) - 4
California
4x2y
Write 10,470,000 in scientific notation. Write 3.4 x 10- 4 in decimal notation. Step 1: Determine the decimal to use to write a number in
scientific notation.
A number in scientific notation has one whole number
with the remaining non-zero digits written after the
decimal point. Drop all zeros after the final non-zero
digit in numbers greater than zero. Drop all zeros
before the first non-zero digit in numbers between 0
and 1.
In 10,470,000, the last non-zero digit when you
read from left to right is 7. Keep zeros between
non-zero digits.
The decimal to use is 1.047.
Step 2: Determine the exponent on 10.
Ask yourself, "How many places would I have
to move the decimal point to get it back to its
original position?"
1~
.
10,470,000 The decimal will move 7 places. Use 7 as the exponent. Since the original number is greater than 0, the exponent is positive. 10,4 70,000 = 1.047 x 107
Please turn the card over for the rest of the problem.
Write 0.00000107 in scientific notation. Write 4.501 x lOG in decimal notation. Step 1: Determine the decimal to use to write a number in
scientific notation.
The first non-zero digit is
The decimal to use is
-----
~~~~-
Step 2: Determine the exponent on 10.
How many places would you have to move the decimal
point to get it back to its original position?
Since the original number is
----
than 0, the
exponent is - - - - - - - 0.00000107
= _ _ _ _ __
Please turn the card over for the rest of the problem.
Write 10,470,000 in scientific notation.
Write 3.4 x 10- 4 in decimal notation.
Step 3: Determine the direction the decimal point moves
when you change a number in scientific notation back
to decimal form.
4
The exponent on 10 in 3.4 x 10- is negative. Move
the decimal point to the left. When the exponent is
positive, move the decimal point to the right
Step 4: Move the decimal point the number of places
indicated by the exponent.
For 3.4 x 10- 4 , move the decimal point 4 places
to the left.
3.4
X
10-
4
3.4 x 10- 4 = 0.00034
© 2002, Renaissance Learning, Inc.
•
Write 0.00000107 in scientific notation. Write 4.501 x 10ri in decimal notation. Step 3: Determine the direction the decimal point moves
when you change a number in scientific notation back
to decimal form.
The exponent on 10 in 4.501 x 106 is
Move the decimal point to the
------
--------
Step 4: Move the decimal point the number of places
indicated by the exponent.
For 4.501 x 106 , move the decimal point
O,gqq~.4
4.501
X
6
10 =
----------------
California
4x2y
Simplify and express in scientific notation.
(1.4 x
10 4)
Simplify and express in scientific notation.
(7.4 x 10
(7.6 x 10 - 3) 2
)
(5.0
X
10
3
)
(8.0 x 10 - 4) (2.5 x 10 - 2) Step 1: Use the commutative property of multiplication to
simplify the numerator.
This helps you put like numbers together.
(1.4 x 10 4) (7.6 x 10- 3) = (1.4 x 7.6) . (10 4 x 10- 3) Step 1: Use the commutative property of multiplication to
simplify the numerator.
Step 2: Use the product of powers property. 104 x 10 - 3 = 10(4 + -3) = 10 1 Step 2: Use the product of powers property.
Multiply the decimals as you normally would.
Multiply the decimals as you normally would.
1
(1.4 x 7.6) x (10 4 x 10 - 3 ) = 10.64 x 10
Step 3: Use the quotient of powers property to simplify
the fraction.
Wi --:- 10- 2 = 10(1 - - 2) = 10 3 Step 3: Use the quotient of powers property to simplify
the fraction.
Divide the decimals as you normally would.
Divide the decimals as you normally would.
10.64 x 101 = 4.256 X 103 2.5 X 10 ­ 2
4
Answer:
(1.4 X 10 ) (7.6 x 10 ~3)
(2.5 x 10 2) © 2002 , Renaissance Learning, Inc. = 4.256
X
10 3 Answer:
(7.4
X
2
10 ) (5.0 x 10:
(8.0 x 10
1
)
-4
)
California
4x2y Ernrn
Last year a large trucking company delivered about 0.8 million
tons of goods at an average value of $25,100 per ton. What was
the total value of goods delivered? Express your answer in
scientific notation.
Step 1: Write numbers in scientific notation.
1 million = 1,000,000
0.8 million = 800,000
4
25,100 = 2.51 x 10
8
=
X 10
Step 1: Write numbers in scientific notation.
2.7 million = ______________
5
0.4 million =
Step 2: Write the problem to be solved.
4
(8 x 2.51) x (10 x 10 )
9
= 20.08 X 10
This number is not in scientific notation since the
number in front of the decimal is greater than nine.
20.08 x 10 9 = 2.008 X 1010
Step 4:
X
10
5
Which operation is needed?
Step 3: Use what you know about exponents to compute.
Step 3: Use what you know about exponents to compute.
4
_ _ _ _ _ _ _ _ _ _ _ _ __
Step 2: Write the problem to be solved.
Describe the problem with smaller numbers to help
you determine which operation to use.
Suppose the problem is about delivering 2 truckloads
of goods worth $500 each.
2 truckloads at $500 each = 2 x $500 = $1000
Use multiplication to solve this problem.
5
4
(8 x 10 ) x (2.51 X 10 )
(8 x 10~ x (2.51
Ms. Z, a pop singer, released a new CD in November. Sales were 2.7 million. In December, sales decreased to 0.4 million. How many times more sales were made in November than in December? )
=
Answer the question.
The trucking company delivered goods worth
. $2.008 x 1010.
© 200 2, Renaissance Learning. Inc.
Step 4: Answer the question.
California