Properties of Exponents and Scientific Notation

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Test
Write the following in either standard form or scientific notation.
1. 0.000006824
2. 3.71 x 100
Simplify the following. Write your answer in both scientific notation and
standard form.
3.
7.1 x 105 – 3.26 x 105
10/30/14
 Exponential Form: 45 x 43
or
59
56
 Expanded Form:
45 x 43 = (4x4x4x4x4)(4x4x4)
59
56
=
5∗5∗5∗5∗5∗5∗5∗5∗5
5∗5∗5∗5∗5∗5
 Standard Form:
45 x 43 = 65,536
59
56
= 625
 RULE: When multiplying two exponential expressions with
the same base, you ADD the exponents.
Example: 59 x 53 = 59+3 =512
The bases are the same (5), therefore you add the two
exponents (9+3).
 RULE: When dividing two exponential expressions with the
same base, you SUBTRACT the exponents.
Example:
38
36
= 38-6 = 32
The bases are the same (3), therefore, you can subtract the
exponents (8-6).
 When DIVIDING two exponential expressions that will result in an
expression with a negative exponent, you have two options:
1. Divide by subtracting the exponents.
43
48
=
_8
3
4
=
_5
4
2. Write the numerator and denominator in expanded form and
simplify.
43
48
=
4∗4∗4
4∗4∗4∗4∗4∗4∗4∗4
=
1
45
1
= 5
4
HINT: You must know your rules for operations with integers in order
to be able to successfully solve problems with negative exponents!
_
45
 Anything raised to the power of zero (0) is always one (1).
70 = 1
• Anything raised to the power of one (1) is always itself.
71 = 7
 When an exponential expression is raised to a power, you
multiply the two exponents.
(88)6 = 88x6 =848
(1011)2 =1011x2 =1022
 Addition (when the signs are the same)
1. Keep the Sign
2. Add
(-9) + (-4) =
 Addition (when the signs are different)
1. Keep the sign of the number with the greater absolute value.
2. Subtract the bigger number from the smaller number.
(-7) + 4 =
 Subtraction (Think L-C-O…Leave, Change, Opposite)
1. Change the subtraction sign to addition sign.
2. Change the sign of the second number.
3. Follow Addition Rules.
 Multiplication/Division
1. Positive & Positive = Positive
2. Negative & Negative = Positive
3. Positive & Negative = Negative
4. Negative & Positive = Negative
8 – 11 =
15 – (-7) =
(-5) x (-9) =
(-90) ÷ 3=
 Scientific Notation: a way to write a number as a
product of the number, a, and 10n, when 1≤ a <10 (a
needs to be at least equal to 1 but less than 10) and n is
an integer.
a x 10n
5.21 x 106
 Standard Form: a way to write a number using a digit
for each place.
591,157.21
5.12 x 106
Step 1: Simplify 106
106 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
Step 2: Multiply by 5.12
5.12 x 1,000,000 = 5,120,000
Hint: The exponent tells you how many spaces to move the
decimal. When converting from scientific notation to standard
form and the exponent is positive, you move the decimal to
the RIGHT and fill spaces with zeroes.
860,000
Step 1:
Step 2:
Identify the location of the decimal point in
860,000. In all whole numbers, the decimal point is
at the end (all the way to the right) of a number.
860,000.
Decimal Point
Move the decimal point to the left until you have a
number that is greater than or equal to 1 and less
than 10. Count the number of places you moved the
decimal point.
860,000.  the decimal is moved 5 places
Step 3:
Rewrite the number in scientific notation. The
number of places you moved the decimal is the
exponent for the base of 10.
8.6 x 105
Remember: Scientific Notation requires that the value for “a” be at least 1
and less than 10.
 Power of Zero
4.5 x 100 = 4.5 x 1 = 4.5
 Negative Exponents
Standard form to Scientific notation: 0.00000651 = 6.51 x 10-6
Count the number of paces the decimal is moved to the right to make the
number between 1 and 10, the number of places moved to the right is
written as the exponent for 10 and should be negative.
Scientific Notation to Standard form: 8.75 x 10-7 = 0.000000875
Move the decimal to the left according to the number in the exponent.
 Ordering
 Use the values of the exponents to help determine the order. The smaller the exponent, the smaller the
value. The greater the exponent, the greater the value.
 Write the number in standard form to check the order.
 Write the numbers in order in their original form (scientific notation).
 To estimate:
1.
2.
Look for the greatest place value and round to that place value.
Follow the rules for converting from standard form to scientific notation.
Examples:
Step 1:
Step 2:
Solution:
0.00005146
456,145,956
(2.15 x 108) x (1.24 x 103)
 Step 1: use the commutative and
associative properties of
multiplication to regroup and
reorder the multiplication problem.
 Step 2: Multiply
Step 1:
(2.15 x 1.24) x (108 x 103)
Step 2:
2.15
108+3 =1011
x1.24
 Solution:
2.666 x 1011
Standard form: 266,600,000,000
16.4 x 109
4.1 x 105
 Step 1: Divide the factors of a.
a x 10n
a
a x 10n
a
 Step 2: Then, apply the rules for
dividing exponential expressions.
(you subtract the exponents)
Step 1:
16.4 = 4
4.1
Step 2:
109-5 =104
n-n
 Solution: Rewrite the answer in
scientific notation using the
number in step 1 and 2.
4 x 104
Standard form: 40,000
Examples:
 Step 1: To add or subtract in scientific notation, the
exponents must be the same. If they are not the same,
rewrite the terms so that the exponents are the same.
To do so, determine the number by which to increase
the smaller exponent by so it is equal to the larger
exponent.
Increase the smaller exponent by this number and
move the decimal point of the number with the
smaller exponent to the left the same number of
places.
4.2 x 103 + 2.9 x 103  same exponents
 Step 2: Add or subtract the new factors (digits) for “a.”
4.2 + 2.9 = 7.1
 Solution: Write the sum in scientific notation. If the
answer is not in scientific notation (i.e. if “a” is not
between 1 and10 ) convert it to scientific notation.
7.1 x 103
8.5 x 108 – 6.2 x 106
2.1 x 104 + 3.6 x 105
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