Algebra (Period 1)

Scientific Notation 5-4



You've had this since 6th grade!
It is just a bit more complicated


You restate very big or very small numbers using powers of 10 in exponential form


Move the decimal so the number fits in this range:

less than 10 and greater than or equal to 1
That is 1 ≤ n < 10 Scientific notation is the product of two factors: one of the factors is a number (1)1 ≤ n < 10 and the other factor (2) is a power of ten 
Count the number of places you moved the decimal and make that your exponent 
Very big numbers - exponent is positive 
Very small numbers (decimals) - exponent is negative (just like a fraction!) 

Remember that STANDARD notation is what you expect (the normal number)
 
When you multiply or divide scientific notations, use the power rules! 
 Just be careful that if your answer does not fit the scientific notation range, that you restate it. (2.2 x 10 ^-3)(3.0 X 10 ⁵)
(2.2)(3.0) X 10 ^-310 ⁵

6.6 X 10 ²

(6.1 x 10 ⁹)(2.5 x 10^-4)
(6.1)(2.5) x 10 ⁹⁺⁴
15.25 x 10 ⁵
But that isn't in scientific notation so change just 15.25 into proper scientific notation first
that is
15.25 = 1.525 x 10¹
put that back in your work
1.525 x 10¹ ⋅ 10⁵
1.525 X 10⁶


(2.5 x 10 ^-3)(4.0 X 10 ^-8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 ^-3)(10 ^-8)

(2.5)(4)= 10
so initially you have 10 X 10 ^-11
but 10 does not fall into the required 1 ≤ n < 10 so changed 10 into scientific notation or 1.0 X 10¹
so you have 1.0 X 10¹ X 10 ^-11
or 1 X 10 ^-10

(5.4 X 10^-6)(5.1 X 10 ^-8)

(5.4)(5.1) = 27.54
so
27.54 X 10 ^-6+-8
27.54 X 10 ^-14
but again 27.54 isn't in scientific notation... changed just that number into scientific notation... FIRST
2.754 X 10¹
put it back into your work
2.754 x 10¹ ⋅ 10^-14
2.754 X 10 ^{1 + -14}
2.754 X 10 ^-13


What about
9.0 X 10 ⁸
3.0 X 10 ²

do your factors separately just like you did with multiplication
That is,
9.0 = 3
3.0


and
10⁸
10²

= 10 ^8-2
=10⁶
so the answer is
3 x 10 ⁶

6.9 X 10 ⁴
3.0 X 10 ⁸

2.3 X 10 ^4-8
2.3 X 10 ^-4

2.7 X 10 ¹²
9.0 X 10 ¹²
Divide 2.7 by 9
you get
0.3
10^{12 -12} = 10 ⁰
but wait 0.3 isn't in the correct for so change that..
0.3= 3.0 x 10^-1
so the answer is
3.0 X 10 ^-1





6.0 x 10 ⁷
3.0 X 10²




First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 ^{7 -2} or 10⁵
so 2.0 X 10⁵

4.2 X 10⁵
2.1 X 10³

= 2 X 10²

2.5 X 10 ^-7
5.0 X 10 ⁶
first 2.5/5 = 0.5 and 10^-7/10⁶ = 10^-7-6 = 10^-13
so initially you have
0.5 x 10^-13 but 0.5 is not within 1 ≤ n < 10 so change 0.5 to scientific notation 5.0 X 10 ^-1 and mult that by 10^-13
5 X 10 ^-14


How about this harder one:

(3.6 X 10⁶)(4 X 10 ^-3
(4.8 x 10 ^-2)(1.2 X 10 ⁶)

put all your factors together and see if you can simplify before you begin...

(3.6)(4)
(4.8)(1.2)

You can simplify
(3.6)
(4.8)


isn't that 3/4
and then your 4's simplify
left with
3.o
1.2
which simplifies to 2.5 ... so going back to the original problem you now have
2.5 x 10^6-3
10 ^{-2 + 6}
which
becomes
2.5 X 10 ^3-4 or
2.5 X 10 ^-1