Which of the following is the best approximation for 11^11 - 9^9 ?

Question

Which of the following is the best approximation for  11^{11}-9^9 ? 
 
A.  11^8  
B.  11^9  
C.  11^{10}  
D.  11^{11}  
E.  11^{12}

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Bunuel · Accepted Answer

ThatDudeKnows · Answer

Given that the section is called Quantitative REASONING, let's get the right answer without doing the math. 9^9 is tiny compared to 11^11. Less than 1%, since 11^11 is more than 11*11*(9^9). We surely are not subtracting enough to decrease by a power of 11.

Answer choice D.

khumoyuns · Answer

11^9 is 121 times greater than 9^9, which is already much smaller than 11^9. Straight away D

KarishmaB · Answer

As explained by everyone above,  9^9  is much smaller than  11^{11}  and that is why subtracting  9^9  doesn't have much impact on the value of  11^{11} .

And if I had to prove it in this question, I would say, ok let me find out whether  11^{11}-9^9  is closer to  11^{11}  or  11^{10}  (the next smaller option)

11^{11} = 11 * 11^{10} = 121 * 11^9

So the ratio of  11^9 : 11^{10} : 11^{11} = 1 : 11 : 121 
Now if I remove 1 from 121, is the result 120 closer to 11 or 121? Obviously 121. 
So if I remove  11^9  from  11^{11} , the result will be much closer to  11^{11}  than to  11^{10} .
Then if I were to remove an even smaller number  9^9  from  11^{11} , the result will obviously be much closer to  11^{11}  than to  11^{10} .

Hence the answer is  11^{11}  i.e. (D)

Usernamevisible · Answer

Why is (9^9) treated as “tiny” compared to (11^11)?

Core idea:
Don’t think “Is 9^9 big?”
Think → “Is 9^9 big relative to 11^11?”

Step 1 — Compare using ratio

(11^11) / (9^9) = (11/9)^9 × 11^2

Approximate:
11/9 ≈ 1.2
1.2^9 ≈ 5
11^2 = 121

So:
ratio ≈ 5 × 121 ≈ 600

Step 2 — GMAT intuition

Subtracting a very small fraction:
600 − 1 ≈ 600

So:
11^11 − 9^9 ≈ 11^11

Key Rule (store this):
If one term is hundreds of times smaller, subtraction ≈ no change

Contrast (important):

Cannot ignore:
11^11 − 10^11
Ratio ≈ 3 → not small

Which of the following is the best approximation for 11^11 - 9^9 ?

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Which of the following is the best approximation for 11^11 - 9^9 ?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards