Chapter R. Section 2
Page 1 of 2
Section R.2 – Integer Exponents, Scientific Notation, Order of Operations
Exponents:
• For any positive integer n, a n = a ⋅ a ⋅ a " a (n times) . a is called the base and n is the exponent
• When you have a negative exponent, swap the term across the fraction bar (if in the numerator move
it to the denominator, and if in the denominator move to the numerator)
1
1
i.e. a − m = m , − n = a n
a
a
• Anything (except 0) to the power of 0 is 1 (a0 = 1)
• Properties of exponents:
a m ⋅ a n = a m+ n
(a m ) n = a mn
(ab) m = a mb m
•
Using the above properties, what would be the following?
•
Example. Simplify a 0 ⋅ a 4
a 0 ⋅ a 4 = a 0+ 4 = a 4
Example. Simplify (−6b −4 )(2b −7 )
•
am ⎛ a ⎞
, ⎜ ⎟
an ⎝ b ⎠
m
−12b −4−7 = −12b −11
•
⎛ 125 p12 q −14 r 22 ⎞
Example. Simplify ⎜
8 6 −15 ⎟
⎝ 25 p q r
⎠
−4
−4
4
⎛ 5 p 4 r 37 ⎞
⎛ q 20 ⎞
q80
=
=
⎜ 20 ⎟
⎜ 4 37 ⎟
16 148
⎝ q
⎠
⎝ 5 p r ⎠ 625 p r
Scientific Notation:
• Scientific notation for a number is an expression of the form N ×10m , where 1 ≤ N < 10 , N is in
decimal form, and m is an integer
• Large numbers have a positive exponent, and small numbers have a negative exponent. It has
nothing to do with the sign of the number, only its size
i.e. 4.5 ×104 = 45, 000 4.5 × 10−4 = 0.00045
• To convert to scientific notation, move the decimal place to the left (or right) and keep track of this
value, until you reach a value of N between 1 and 10, this will be +m (moving left) or –m (moving
right)
• Example. Convert 1,670,000 to scientific notation
1670000 = 1.67 ×106
• Example. Convert 0.00000000514 to scientific notation
5.14 ×10−9
• To convert from scientific notation, write down N, move the decimal place m units (to the left if
negative, to the right if positive)
C. Bellomo, revised 26-Aug-07
Chapter R. Section 2
Page 2 of 2
•
•
•
•
Example. Convert 3.15 ×10−6 to decimal notation
0.00000315
Example. Convert 8.409 ×1011 to decimal notation
840,900,000,000
When computing with scientific notation, all the rules for exponents apply
1.3 × 104
Example. Compute
5.2 × 1010
1.3
× 10−6 = 0.25 × 10−6 = 2.5 × 10−7
5.2
Order of Operations:
• It is essential to do computations in the proper order, or the result will be incorrect
• Groupings, exponents, multiplication/division, addition/subtraction
• Example, page 14 number 88. Calculate 3[(2 + 4 ⋅ 22 ) − 6(3 − 1)]
3[(2 + 4 ⋅ 4) − 6(3 − 1)] = 3[(2 + 16) − 6(2)] = 3[18 − 12] = 3[6] = 18
• Be sure you are familiar with how your calculator works as well, as many operate differently
C. Bellomo, revised 26-Aug-07