If n is an integer, is (0.1)^n greater than (10)^n?

Question : is (0.1)^n > (10)^n?
Question : is (1/10)^n > (10)^n?
Question : is (10)^{-n} > (10)^n?
Question : is (-n) > n?
Question : is (2n) < 0?

Question : is n < 0?



Statement 1: n > −10
n may be Negative or Positive. Hence
NOT SUFFICIENT

Statement 2: n < 10
n may be Negative or Positive. Hence
NOT SUFFICIENT

Combining the two statements
even after combining the two
-10 < n < 10
i.e. n may be Negative or Positive. Hence
NOT SUFFICIENT

Answer: Option E

Is (0.1)^n > (10)^n ---> \(10^{-n} > 10^n\) ---> \(10^{2n} < 1\) ---> \(10^{2n} < 10^0\) ---> 2n < 0 ---> is n<0?

Per statement 1, n>-10, yes for n=-5 but no for n=5. Not sufficient.

Per statement 2, n<10, yes for n=-5 but no for n=5. Not sufficient.

Combining, the above 2 cases still apply giving you an ambiguous answer.

Thus, E is the correct answer.
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Target question: Is (0.1)^n greater than (10)^n?

REPHRASED target question: Is (1/10)^n greater than (10)^n?

Statement 1: n > −10
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of n that satisfy statement 1. Here are two:
Case a: n = 1, in which case (1/10)^n is NOT greater than (10)^n
Case b: n = -1, in which case (1/10)^-1 = 10 and 10^-1 = 1/10. Here, (1/10)^n IS greater than (10)^n
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: n < 10
There are several values of n that satisfy statement 2. Here are two:
Case a: n = 1, in which case (1/10)^n is NOT greater than (10)^n
Case b: n = -1, in which case (1/10)^-1 = 10 and 10^-1 = 1/10. Here, (1/10)^n IS greater than (10)^n
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

IMPORTANT - Notice that I tested the SAME VALUES for both statements. This means that, the STATEMENTS COMBINED are also NOT SUFFICIENT

Answer = E

Cheers,
Brent

If n is an integer, is (0.1)^n greater than (10)^n?

(1) n > −10
(2) n < 10

Sol. (10)^-n > (10)^n or -n>n or n<0 ?
1) n > -10 Not sufficient
2) n < 10 Not sufficient

1) + 2) -10 < n <10 Not sufficient

Hence E

Please note that for n=0, both the expressions will be equal to 1.

Hence, Answer: E.

Here's another approach:

Target question: Is (0.1)^n > (10)^n?
This is a good candidate for rephrasing the target question.

Since (0.1)^n is always POSITIVE, we can safely divide both sides of the inequality by (0.1)^n to get: 1 > [(10)^n]/[(0.1)^n]
There's a nice rule that says (a^n)/(b^n) = (a/b)^n
When we apply this rule to the right side of the inequality, we get: 1 > (10/0.1)^n
Simplify to get: Is 1 > 100^n?
Notice that, when n = 0, then 100^n = 100^0 = 1
So, when n > 0, then 100^n > 1, and when n < 0, then 100^n < 1
So, we can REPHRASE the target question as....
REPHRASED target question: Is n < 0?

Statement 1: n > -10
There are several values of n that satisfy statement 1. Here are two:
Case a: n = -9, in which case n < 0
Case b: n = 2, in which case n > 0
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n < 10
There are several values of n that satisfy statement 1. Here are two:
Case a: n = -9, in which case n < 0
Case b: n = 2, in which case n > 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer:

RELATED VIDEOS

Bunuel can we play around with the statement in question like this?
I mean to say:

Its given that is 10^-n>10^n

So can we divide/subtract/multiple/add on both sides of an expression that is to be proved?

When should we not do it?


Thanks

How to manipulate inequalities (adding, subtracting, squaring etc.).
_________________

Video solution from Quant Reasoning:
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Solution:

Question Stem Analysis:

We need to determine whether 0.1^n is greater than 10^n, given that n is an integer. Notice that 0.1 = 10^(-1); therefore, 0.1^n = 10^(-n) and 10^(-n) is greater than 10^n if -n > n. We see that -n > n if and only if n is negative. In other words, we need to determine whether n is negative.

Statement One Alone:

Even though we know n > -10, n could be either positive or negative. So we cannot definitely say n is negative. Statement alone is not sufficient.

Statement Two Alone:

Even though we know n < 10, n could be either positive or negative. So we cannot definitely say n is negative. Statement two is not sufficient.

Statements One and Two Together:

With the two statements, we see that -10 < n < 10. Thus, n could still be either positive or negative. Both statements together are not sufficient.

Answer: E
_________________

Answer: Option E

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IMO here is the simplest explanation:

Let’s analyze the question --------- the only case when 0.1^n > 10^n is if n is negative
So basically, we are trying to find if n < 0

Statement 1:
N > -10
Here n can be ------- n < 0 or if the value goes beyond 0 then n > 0
NOT SUFFICIENT

Statement 2:
N < 10
Here n can be --------- n > 0 or n < 0
NOT SUFFICIENT

Statement 1+2
-10 < n < 10
Again n can be ------- n < 0 or n > 0
NOT SUFFICIENT

Answer – E