Download The Power of Exponents

The Power of Exponents.
Exponents are good for much more than providing one more math topic for
struggling students. As an example, consider representing numbers of very
different sizes. Here, exponents will be used to represent place value.
Sizes of things, in meters.
Bacterium
0.000001 m
Hydrogen atom
0.00000000005 m
Milky Way Galaxy 1000000000000000000000 m
Distance to Nearest Star 40000000000000000 m
Atomic Nucleus
0.000000000000005 m
Chemical Bond
0.000000004 m
Earth
1280000 m
It’s easy to get lost in all those zeroes! (Was that 15 or 17 zeroes?) There
is a much simpler way to write these numbers using powers of ten. In
scientific notation, a number is written as a number between 1 and 10
multiplied by ten to a power, like 5.4 x 106
Example:
The speed of light is 300000000 meters per second (m/s).
That’s 3 with 8 zeroes after it.
With 8 zeroes after the three, we need to multiply the 3 by 100000000.
100000000 = 10x10 x10 x10 x10 x10 x10 x10 = 108
so 300000000 = 3.0 x 108
Calculating with exponents.
Multiplication: (5 x 105)(3 x 102) = 15 x 105+2 = 15 x 107 or 1.5 x 108
Division:
6 x 108 / (2 x 103) = 3 x 108-3 = 3 x 105
More involved: F =(6 x 10-11)(3 x 1027)(4 x 1032)/(6 x 109)2
Separate numbers from powers of 10:
F = [(6 x 3 x 4)/62] x [ (10-11)( 1027)( 1032)/( 109)2]
=
2
x [ 10(27+32-11)/10(2x9)]
= 2 x 10(48-18) = 2 x 1030
Powers and inverse function, the log.
Just as we use exponents to simplify our numbers, we also use them as a
function. For example,
Y = 10x ; for x = 1.301, y = 101.30 = 20, to 4 figures.
This also leads to the inverse of this function, the log.
If Y = log(x), then 10y = 10log(x) = x .
Just as division undoes multiplication, logs undo power functions. We use 2
log functions in science, with bases 10 and e:
y = log(x) means x = 10y
y = ln(x) means x = ey
The latter is referred to as the natural log (hence the ln for log natural)
because it shows up so often in nature.
Some rules of calculation with logs:
log(a+b) = log(a+b), no simplification.
log(ab) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(bp) = p log(b)
Bad joke of the day:
A zoo had a pair of rare snakes that it was trying to breed. They had not had any luck for
months and moved the snakes to a new enclosure which happened to have a table made of
rough cut logs. Within a month young snakes were wriggling about on the table. The zoo
keepers learned an important lesson:
With log tables, even adders can multiply!