Is x > 3^10 ?

[kevincanspain]

Is x > 3^10 ?

(1) x > 3^12 - 3^4
(2) x > 10^5

[gaurav_gaur]

1) x > 3^12 - 3^4
i.e. x > 3^4*(3^8-1).......(a)

if I look back at the question it states that
is x > 3^10
i.e. x > 3^4 * 3^6.......(b)

By comparing the 2 inequalities a and b, I can infer that if
(3^8-1) > 3^6 then x > 3^10.

Simply by looking at the equation i can say that 3^8 is any day greater than 3^6 and subtracting 1 from 3^8 will not harm the result much. thus,
x > 3^4(3^8 - 1)
and hence x > 3^4 * 3^6

So statement 1 is sufficient.


2)x > 10^5
again, if I look back at the question it states that
is x > 3^10
i.e. x > 9^5

10^5>9^5

so x > 10^5
then x > 9^5 as well.

So statement 2 is sufficient.

OA should be D

Cheers,
GG

[srcc25anu]

Is x > 3^10?

1. x > 3^12 - 3^4
x > 3^4 (3^8 - 1)
x > 3^4 * (3^4 + 1) * (3^4 - 1)
x > 3^4 * (3^4 + 1) * (3^2 + 1) * (3^2 - 1)
x > 3^4 * ~3^4 * ~3^2 * ~3^2
x > ~3^12

Sufficient

2. x > 10^5
3^2 = 9
(3^2)^5 or 3^10 = 9^5

10^5 is greater than 9^5
Sufficient

Ans D

[Anju@Gurome]

1. x > 3^12 - 3^4
x > 3^4 (3^8 - 1)
x > 3^4 * (3^4 + 1) * (3^4 - 1)
x > 3^4 * (3^4 + 1) * (3^2 + 1) * (3^2 - 1)
x > 3^4 * ~3^4 * ~3^2 * ~3^2
x > ~3^12

Although you've got the correct answer, I just want to add that those three lines marked in green are unnecessary time waste. When you can approximate (3^2 - 1) as 3^2, you can definitely approximate (3^8 - 1) as 3^8.

So without doing unnecessary calculations, you can directly approximate (3^4)*(3^8 - 1) as (3^4)*(3^8) = 3^12

Hope that helps.

[Brent@GMATPrepNow]

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

[iamniladri]

I have simplified the Statement - 2 in the following way:
Given : x > 10^5
Question: Is x > 3^10? ( or x > 9^5)
Because 3^10 can be written as 9^5 as 3^10 = (3*3)*(3*3)*(3*3)*(3*3)*(3*3).
Now, if x > 10^5 then it is certain that x > 9^5. So SUFFICIENT.

Thanks.

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

[chrislacolla]

all of the calculations unnecessary in this case. each statement gives you a value of x- regardless of what that value is, it will allow you answer the original yes/no question. it doesn't matter whether it's yes or no, just that you can answer it given whatever the expressions in 1 and 2 simplify to.

[Brent@GMATPrepNow]

all of the calculations unnecessary in this case. each statement gives you a value of x- regardless of what that value is, it will allow you answer the original yes/no question. it doesn't matter whether it's yes or no, just that you can answer it given whatever the expressions in 1 and 2 simplify to.

That would be true if each statement gave us the actual value of x. However, each statement is an inequality (not an equation) and does not give us the value of x.

For example, consider this question.
Is x > 10?
(1) x > 2
(2) x > 3
In this case the each statement does not sufficient.

Conversely, this question analogous to the original question.
Is x > 10?
(1) x > 12
(2) x > 13
Here, each statement is sufficient because each one ensures that x > 10.

So, the given inequalities need to be examined in greater detail than you suggest.

Cheers,
Brent