[Math Revolution GMAT math practice 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
Which of the following is the best approximation for 11^11-9
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- Max@Math Revolution
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11 = 1.1 x 10Max@Math Revolution wrote:[Math Revolution GMAT math practice 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
9 = .9 x 10
Reformulating:
(1.1x10)^11 - (.9 x 10)^9 = 11^11 - 9^9
Splitting out the parts:
(1.1 x 10)^ 11 = 1.1^11 x 10^11 and (.9 x 10)^9 = .9^9 x 10^9
Factoring out the 10^9:
10^9(1.1^11 x 10^2 - ,9^9 x 1)
Now .9^9 is essentially 0, so this now reduces to 10^9(1.1^11 x 10^2) = 1.1^11 x 10^11 = [spoiler] 11^11,D[/spoiler]
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Since the answer choices are very spread out, our approximation does not have to very exact.Max@Math Revolution wrote:[Math Revolution GMAT math practice 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
11¹¹−9�
= 11¹¹− (11−2)�
≈ 11¹¹ − 11�
= 11�(11²−1)
≈ (11�)(11²)
= 11¹¹
The correct answer is D.
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Another approach:Max@Math Revolution wrote:[Math Revolution GMAT math practice 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
10¹¹ (100,000,000,000) is 100 TIMES the value of 10� (1,000,000,000).
Implication:
11¹¹ must be MORE THAN 100 TIMES the value of 9�.
Since 9� is so small when compared to 11¹¹, we can ignore the value of 9� when we approximate:
11¹¹ - 9� ≈ 11¹¹
The correct answer is D.
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- Max@Math Revolution
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=>
9^9 is much smaller than 11^11.
Thus, 11^11-9^9 is closest to 11^11.
Therefore, the answer is D.
Answer: D
9^9 is much smaller than 11^11.
Thus, 11^11-9^9 is closest to 11^11.
Therefore, the answer is D.
Answer: D
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Max@Math Revolution wrote:[Math Revolution GMAT math practice 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
Each term is a large number, however, 9^9 is much smaller than 11^11 when it's compared to 11^11. Therefore, the difference will still be approximately 11^11.
(Note: If you can't see that 9^9 is much smaller than 11^11, let's compare their ratio:
9^9/11^11 < 10^9/10^11= 1/10^2 = 1/100 = 1%
We see that 9^9 is less than 1% of 11^11, that is, their difference, 11^11 - 9^9 will be more than 99% of 11^11.)
Alternate Solution:
Let's approximate 9 to 11:
11^11 - 9^9 ≈ 11^11 - 11^9 = 11^9(11^2 - 1)
We see that 11^2 - 1 is approximately 11^2; therefore 11^11 - 9^9 is approximately 11^9 * 11^2 = 11^11.
Answer: D
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