Download Advanced Seventh Grade First Ten Weeks

What is an exponent
The base a is raised to the power of n is equal to the multiplication of
a, n times:
a n = a × a × ... × a
n times
a is the base and n is the exponent.
Examples
31 = 3
32 = 3 × 3 = 9
33 = 3 × 3 × 3 = 27
Exponents rules and properties
Rule name
Product rules
Quotient rules
Rule
Example
a n · a m = a n+m
23 · 24 = 23+4 = 128
a n · b n = (a · b) n
32 · 42 = (3·4)2 = 144
a n / a m = a n-m
25 / 23 = 25-3 = 4
a n / b n = (a / b) n
43 / 23 = (4/2)3 = 8
(bn)m = bn·m
(23)2 = 23·2 = 64
Power rules
Negative exponents b-n = 1 / bn
Zero rules
One rules
2-3 = 1/23 = 0.125
b0 = 1
50 = 1
0n = 0 , for n>0
05 = 0
b1 = b
51 = 5
1n = 1
15 = 1
Minus one rule
Exponents product rules
Product rule with same base
an · am = an+m
Example:
23 · 24 = 23+4 = 27 = 2·2·2·2·2·2·2 = 128
Product rule with same exponent
an · bn = (a · b)n
Example:
32 · 42 = (3·4)2 = 122 = 12·12 = 144
See: Multplying exponents
Exponents quotient rules
Quotient rule with same base
an / am = an-m
Example:
25 / 23 = 25-3 = 22 = 2·2 = 4
Quotient rule with same exponent
an / bn = (a / b)n
Example:
43 / 23 = (4/2)3 = 23 = 2·2·2 = 8
See: Dividing exponents
Exponents power rules
Power rule I
(an) m = a n·m
Example:
(23)2 = 23·2 = 26 = 2·2·2·2·2·2 = 64
Power rule II
m
m
a n = a (n )
Example:
2
2
23 = 2(3 ) = 2(3·3) = 29 = 2·2·2·2·2·2·2·2·2 = 512
(-1)5 = -1
Negative exponents rule
b-n = 1 / bn
Example:
2-3 = 1/23 = 1/(2·2·2) = 1/8 = 0.125
Guided notes: Scientific notation
When using Scientific Notation, there are two kinds of exponents:
positive and negative.
When changing scientific notation to standard notation, the exponent
tells you if you should move the decimal:
With a positive exponent, move the decimal to the right:
4.08 x 103 = 4 0 8 0 (show arrows and move decimal)
The exponent tells you how many places to move the decimal.
With a negative exponent, move the decimal to the left:
4.08 x 10-3 =
4 0 8
The exponent tells you how many places to move the decimal.
When changing from Standard Notation to Scientific Notation:
1) First, move the decimal after the first whole number:
3 2 5 8. (use arrows to show how you move the
decimal)
2) Second, add your multiplication sign and your base (10).
3 . 2 5 8 x 10
3) Count how many spaces the decimal moved and this is the exponent.
3 . 2 5 8 x 10
(add the correct exponent)
When you multiply in scientific notation, just multiply the decimal
numbers and ADD the exponents
.00000055 x 24,000
= (5.5 x 10-7) x (2.4 x 104)
= (5.5 x 2.4) x 10-7+4
= 13 x 10-3 = 1.3 x 10-2
When you divide in scientific notation, just divide the decimal numbers
and SUBTRACT the exponents
• (7.5 x 10-3)/(2.5 x 10-4)
= 7.5/2.5 x 10-3-(-4)
= 3 x 10 = 30
When adding numbers written in scientific notation:
1. First make sure that the numbers are written in the same form
(have the same exponent)
Ex. 3.2 x 103 + 40 x 102 (change to 4.0 x 103)
2. Add (or subtract) the decimal numbers
3.2 + 4.0 = 7.2
3. The rest of the exponent remains the same
Answer: 7.2 x 103
How do you make the exponents the same?
1. Let’s say you are adding 2.3 x 103 and 2.1 x 105. You can either
make the 103 into the 105 or visa versa. If you make the 103 into 105,
you are moving up the exponent two places. You will need to move your
decimal place in the decimal number down two places to the left.
2. 2.3 x 103 = .023 x 105
• (take 2.3 and move the decimal three places to the right. It
equals 2300.)
• (take .023 and move it five places to the right…it is still 2300)
3. Now add the two mantissas (2.1 + .023) = 2.123
4. Add the exponent ending: 2.123 x 105
In conclusion, If you move the exponent up, you must move the decimal
in the decimal number down to the left the same number of places. If
you move the exponent down, you must move your decimal point to the
right in the decimal number that number of places.
Note: When solving an inequality, if you solve by multiplying or dividing
using a negative number----you must reverse the inequality sign.