kevincanspain wrote:Is x > 3^10 ?
(1) x > 3^12 - 3^4
(2) x > 10^5
Target question:
Is x > 3^10 ?
Statement 1: x > 3^12 - 3^4
We need to compare 3^12 - 3^4 with 3^10.
If 3^12 - 3^4 > 3^10, then we can be certain that x > 3^10
Here's the "number sense" approach.
First, let's ignore the 3^4 for a moment.
Notice that 3^12 is WAYYYY bigger than 3^10
In fact, since 3^12 = (3^2)(3^10), we can see that 3^12 is
9 TIMES the value of 3^10
As such, subtracting 3^4 from 3^12 is going to have little effect on 3^12.
In other words, we can be quite certain that 3^12 - 3^4 > 3^10, which means x > 3^12 - 3^4 > 3^10, in which case we can be certain that
x > 3^10
For those who don't like the number sense approach, here's a quick proof.
First recognize that x > 3^12 - 3^4
> 3^12 - 3^10
Now examine
3^12 - 3^10
3^12 - 3^10 = 3^10(3^2 - 1) = 3^10(8)
Since
3^10(8) > 3^10, we can write:
x > 3^12 - 3^4
> 3^12 - 3^10 > 3^10
From here, we can be certain that
x > 3^10
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: x > 10^5
Since 3^2 = 9 and 4^2 = 16, we can say that (3.something^2) = 10
Now replace 10 with (3.something^2)
We get: x > (3.something^2)^5
Simplify: x > (3.something)^10
Since (3.something)^10 > 3^10, we can write: x > (3.something)^10 > 3^10
From here, we can be certain that
x > 3^10
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer =
D
Cheers,
Brent
