Exponents

In this post we will try to understand some properties of exponents.

How much is 2 ³ ? = 2*2*2 = 8

2 ⁴ = 2*2*2*2 =16
2 ⁵ = 2*2*2*2*2 =32
2 ⁶ = 2*2*2*2*2*2 =64
2 ² = 2*2 =4

Lets put all that in sequence...

2 ² 2 ³ 2 ⁴ 2 ⁵ 2 ⁶
4, 8, 16, 32, 64

Note that when we increase in the exponent we multiply by 2!
So the next terms would be:

128, 256, 512,...

But what the previous terms should be?

When we decrease the exponent... we divide by two!!

2 ^-1 2 ⁰ 2 ¹ 2 ² 2 ³ 2 ⁴ 2 ⁵ 2 ⁶
  z,  y,  x, 4,  8, 16, 32, 64

x = 4/2 =2
y = x/2 =2/2=1
z = y/2 =1/2 =1/2

2 ^-1 2 ⁰ 2 ¹ 2 ² 2 ³ 2 ⁴ 2 ⁵ 2 ⁶
1/2,1,  2,  4,  8, 16,  32,64

In here notice 2 ¹ =2 -that does make some sense
and 2 ⁰ =1!

some more terms:

2 ^-3 2 ^-2 2 ^-1 2 ⁰ 2 ¹ 2 ² 2 ³
  c,  b, 1/2,  1,  2,  4,  8

b = (1/2)/2 = 1/4
c = b/2 = 1/8

1/8,1/4,1/2,1,2,4,8

Notice that the numbers (1,2,4,8) 'repeat' 1/8=2 ^-3 = 1/(2 ³ )

And that is what happens in general with negative exponents.

2 ^-n = 1/(2 ⁿ )

Now some more properties

2 ³ *2 ⁵ = (2*2*2)*(2*2*2*2*2)= 2 "eight times" =2 ⁸

So when we multiply two powers in the same base we add the exponents

For division:

2 ⁷/2 ⁴ = (2*2*2*2*2*2*2)/(2*2*2*2)= 2 "three times" =2 ³

There is some cancellation, so when we divide two powers in the same base we subtract the exponents.

(2 ³ ) ⁴ = 2 ³ 2 ³ 2 ³ 2 ³ = 2 "twelve times=3*4" = 2 ¹²

Kind of strange to put in words (try it!!) but easier to understand.