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

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Tue Aug 21, 2018 2:33 am

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[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

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by regor60 » Tue Aug 21, 2018 7:10 am

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

11 = 1.1 x 10
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]

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Aug 21, 2018 7:23 am

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

Since the answer choices are very spread out, our approximation does not have to very exact.
11¹¹−9�
= 11¹¹− (11−2)�
≈ 11¹¹ − 11�
= 11�(11²−1)
≈ (11�)(11²)
= 11¹¹

The correct answer is D.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Aug 21, 2018 7:48 am

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

Another approach:
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.

Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Thu Aug 23, 2018 1:07 am

=>

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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by Scott@TargetTestPrep » Thu Aug 23, 2018 4:12 pm

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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