Download 2. Exponents and Powers of Ten

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TOPIC 2: EXPONENTS AND POWERS OF TEN
EXPONENTS 2.1: Powers of Ten
The powers of 10 are in a pattern. If you look at the pattern, you can see why
0
10 = 1, and why the negative powers of 10 mean “reciprocal.”
For Example: 10 −1 =
1
1
−2
, 10 = 2 , and so on.
10
10
You can also see why multiplying or dividing by 10 is as simple as moving the decimal point:
103 = 1000.
102 = 100.
101 = 10.
100 = 1.
10 −1 = 0.1 =
1
1
= 1
10 10
10 −2 = 0.01 =
1
1
= 2
100 10
10 –3 = 0.001 =
1
1
= 3
1000 10
To divide by 10, all you have to do is move the decimal point one digit to the left; and
to multiply by 10, all you have to do is move the decimal point one digit to the right.
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 2
This is a very useful shortcut if you are doing computations with powers of 10.
EXPONENTS 2.2: Multiplying Numbers that Have Exponents
To multiply two numbers with the same base, add the exponents.
fs × ft = fs+t
Here f is called the base. The two numbers f s and f t have the same base. So the exponents s and t can be added.
EXAMPLE: 124 × 125 = 129
WHY: (12 × 12 × 12 × 12) × (12 × 12 × 12 × 12 × 12) = 129
The base, 12, is the same throughout, so we can just add the exponents 4 + 5 = 9. The
9
answer is 12 .
EXPONENTS 2.3: Raising Exponents to a Power
To raise an expression that contains a power to a power, multiply the exponents.
v w
v×w
(j ) = j
The exponents v and w are multiplied.
EXAMPLE: (32)3 = 36
WHY: (3 × 3) × (3 × 3) × (3 × 3) = 36
EXPONENTS 2.4: Dividing Numbers that Have Exponents
To divide two numbers that have the same base, subtract the exponents.
am
= am − an
an
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 3
The two numbers am and an have the same base, a. So the exponents m and n can be
subtracted.
EXAMPLE:
WHY:
10 3
= 10 3− 2 = 101 = 10
10 2
10 × 10 × 10
= 10
10 × 10
EXPONENTS 2.5: Multiplying Numbers in Scientific Notation
When you are multiplying, order doesn’t matter. You can choose a convenient order.
(a × b) × (c × d) = (a × c) × (b × d)
2
4
EXAMPLE: Multiply (4.1 × 10 ) × (3.2 × 10 ). Rearrange the order of the factors.
Group the powers of 10 together.
2
4
6
(4.1 × 3.2) × (10 × 10 ) = 13 × 10
This answer could also be written as 1.3 × 107. However, in science, we’re fond of powers of three, so there are prefixes that can substitute for powers that are multiples of
three. If, for example, the above number were a length in nanometers, it could be written as 1.3 × 107 nm or as 13 mm. This conversion is legitimate because 1.3 × 107 nm = 13
6
6
7
× 10 nm, and 10 nm = 1 mm, so 1.3 × 10 nm = 13 mm. (See the discussion on signifi6
6
cant digits in topic 3 to see why the answer is 13 × 10 nm and not 13.12 × 10 nm.)
In scientific notation, the number multiplying the power of 10 is called the mantissa.
6
In 5 × 10 , 5 is the mantissa. When there is no mantissa written, the mantissa is 1. So
6
6
6
10 has a mantissa of 1: 10 = 1 × 10 .
5.678 × 10 3
= 5.678 × 10 −3
EXAMPLE:
6
10
6
6
To see why, write 10 as 1 × 10 and rearrange the factors:
EXAMPLE:
5.678 × 10 3 5.678 10 3
=
× 6 = 5.678 × 10 −3
6
1 × 10
1
10
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 4
In scientific notation, when no power of 10 is written, the power of 10 that is meant
is 100. That’s because 100 = 1.
EXAMPLE:
5.678 × 10 3
= 1.38 × 10 3
4.12
0
To see why, write 4.12 in scientific notation as 4.12 × 10 . Then rearrange the factors
and divide.
5.678 × 10 3 5.678 10 3
=
× 0 = 1.38 × 10 3
0
4.12 × 10
4.12 10
A word about notation:
Another way of writing scientific notation uses a capital E for the × 10. You may
have seen this notation from a computer.
EXAMPLES: 4.356 × 107 is sometimes written as 4.356E7.
2.516E5 means 2.516 × 105.
The letter E is also sometimes written as a lowercase e; this choice is unfortunate
because e has two other meanings.
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 5
EXPONENTS – Try It Out
EXERCISE I:
Solve the following equations; all answers should be in the form of a
number (or variable) raised to an exponent:
EXAMPLE: 144 × 148 = 1412
A.
6
1. 10 × 104 =
2. 103 × 103 =
3. 10123 × 100 =
4. 104 × 104 =
5. 10n × 10m =
6. 33 × 32 =
7. n4 × n12 =
8. h7 × h7 =
9. an × am =
10. j 3 × j m =
B.
3 3
1. (6 ) =
2. (62)3 =
3. (104)2 =
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 6
4. (106)3 =
5. (22)2 =
2 10
6. (16 ) =
7. (k3)4 =
8. (m6)2 =
9. (nk)l =
10. (sm)2 =
C.
1.
10 6
=
101
2.
10 2
=
10 4
3.
10
10
1
2
1
4
=
10 m
=
4.
10 n
5.
m10
=
m7
mt
=
6.
m ( t −1)
( n + m )5
=
7.
( n + m )2
8.
j (n+4)
=
j ( n−6)
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9.
e 2π
=
eπ
10.
qj
=
qh
TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 7
D.
1. (3 × 106) × (4 × 104) =
2. (5 × 107) × (7 × 10–2) =
3. (1.6 × 102) × (4 × 103) =
4. (−4.6 × 106) × (−2.1 ) × 104 =
5. (1.11 × 100) × (6.00 × 101) =
LINKS TO ANSWERS
EXERCISE I A.
EXERCISE I B.
EXERCISE I C.
EXERCISE I D.
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 8
TRY IT OUT: ANSWERS
A.
10
1. 10
2. 106
3. 10123
4. 108
5. 10(n+m)
6. 35
7. n16
8. h14
9. a (n+m)
10. j (3+m)
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 9
TRY IT OUT: ANSWERS
B.
9
1. 6
2. 66
3. 108
4. 1018
5. 24
6. 1620
7. k12
8. m12
9. nkl
10. s2m
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 10
TRY IT OUT: ANSWERS
C.
1. 105
2. 10−2
3. 10
1
4
4. 10(m − n)
5. m3
6. m1 or m
7. (n + m)3
8. j 10
9. eπ
10. q (j − h)
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TOPIC 2: EXPONENTS AND POWERS OF TEN / Page 11
TRY IT OUT: ANSWERS
D.
1. 12 × 10
10
2. 35 × 105
3. 6.4 × 105
4. 9.66 × 1010
5. 6.66 × 101
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