Sunday, January 26, 2014

7: Exponents.

What's bigger?  \$2^{100} or {100}^2\$ ?

1. This one is nice if you do a little playing with laws of exponents, factoring, and reducing the problem to numbers you can calculate easily by hand or even in your head.

2^100 = 2^4(2^96) [the motivation for the choice of factoring here will be obvious when we look at the left side]

100^2 = (2*50)^2 = 2^2 * 50^2
= 2^2 * (2^25) ^2
= 2^2 * 2^2 * 25^2
= 2^4 * 25^2 (see above)

= 2^4 * 625

so now we have 2^4 * 2^96 compared with 2^4 * 625
Dividing both sides by 2^4 we are left with comparing 2^96 with 625

Taking some powers of 2, we discover that 2^10 = 1024 which is clearly greater than 625.

So there is no need to know the value of 2^96, or 2^100 (or even of 100^2, though that's an easy mental calculation).

2. Following up on Michael Paul Goldenberg's comment, if we allow for the possibility that 2^10 = 1024 has been discovered, then we only have to observe that 2^100 = (2^10)^10 = (1024)^10 > (1000)^10 = (10^3)^10 = 10^30. That is, 2^100 is greater than 10^30, and of course 10^30 is greater than 100^2 = 10^4, so it follows that 2^100 is (much) greater than 100^2.