Question: Is \(5^n < 0.04\)?

1. Bonnie and Clyde complete the painting of the car at 10:30 am
2. \(x^2+y^2<12\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Approach Solution (1):
Notice that for \(n\geq0\)so RHS of the Q stem is always greater than zero so we need to confirm whether n < -2 because \(5^{-2} is {1\over25} =0.04\)

S1 says \((\frac{1}{5})^n >25\) or n > 2
So S1 is sufficient as n is positive
S2 says \(n^3 < n^2 or n (n^2 -1) < 0 or n (n-1)(n+1) <0\)

Key points are n = 0, -1 and 1
So we have 4 ranges:
Case 1: n < -1, the expression n ( n – 1) ( n + 1) < o is true
Case 2: -1 < n < 0, the expression is false as it will be > 0
Case 3: 0 < n < 1, the expression will be true
And
Case 4: n > 1, the expression will be true for all values
Since, we have 2 ranges in which expression holds true and for those range n < -2 or n > -2

Correct option: A

Approach Solution (2):
Given:
\(5^n < 0.04\)
\(5^n < 0.04\)\((0.04={4\over100}={1\over25}=5^{-2})\)
\(n<-2?\)
Option A:
\(5^{-n}>25=5^2; \) implies n < -2
Sufficient
Option B:

\(n^3 < n^2\)
\(\Rightarrow n^3 -n^2 <0\)
\(\Rightarrow n^2(n-1)<0\)
\(\Rightarrow n-1<0\)
\(n < 1\)
Not sufficient

Correct option: A

Approach Solution (3):
S1:

\(\frac{1}{5}^n > 25\)
\(5^{-n} > 25\)
\(5^{-n} > 5^2\)
\(-n > 2\)
\(n < -2\)

Plug in n = -3 into the equation will yield that \(1\over125\)is < than 0,04
Sufficient
S2:
n can be either a negative number or a proper fraction
\((5)^{1\over2}\)would be > 0,04 while to a negative power would be <.
Not sufficient

Correct option: A

“Is \(5^n < 0.04\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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