Any simple way to find 9^19 - 10^18?

Question

without using calculator? I don't need a precise answer. Options of answer are given. The answer I'm supposed to get is 3.5 x 10^17

No Binomial expansion, since it's not a simple way in this case because of the high power and factorials (can't use calculator).

Answer 1

Start with:

9^18 / 10^18 = 0.9^18 = 0.15 = 1.35/9

Cross multiply:

9^19=1.35x10^18=(1 + 0.35 )x10^18=10^18+0.35 x10^18

=10^18+3.5x10^17

This leads to:

9^19 - 10^18 = 3.5x10^17

Answer 2

I only starred this question to see how someone else would do it. But no luck so far. Here's what I have so far.

9^19 - 10^18 = 10^18 * [9*(9/10)^18 - 1] ...(1)

Now let's try to estimate (9/10)^18 = (1-0.1)^18
y = (1-0.1)^18
lny = 18*ln(1-0.1)
=~ -18*(0.1+0.1^2/2 + 0.1^3/3)
= -1.896
This can easily be worked out manually
y = e^(-1.896)

We know e^2 =~ 7.something
So we have to estimate e^1.896
I'd guess ~ 7
So y =~ 1/7

Because of the error in estimating y, it gives us a final answer of 2.9 x 10^17

Hopefully someone can come up with a better solution before the question expires.

*EDIT*
I'm not sure how the answerer below got 0.9^18 = 0.15 without using a calculator.