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26 Jan 2024, 02:18
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
86% (02:14) correct
14%
(02:36)
wrong
based on 526
sessions
History
Date
Time
Result
Not Attempted Yet
If \(3^x - 3^{x - 3} = 2,106\), what is the value of x?
A. 10
B. 9
C. 8
D. 7
E. 6
2024-01-26_13-12-27.png [ 13.13 KiB | Viewed 11420 times ]
A. 10
B. 9
C. 8
D. 7
E. 6
Attachment:
2024-01-26_13-12-27.png [ 13.13 KiB | Viewed 11420 times ]
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26 Jan 2024, 02:23
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guddo
Factor out \(3^{x - 3}\):
- \(3^{x - 3}(3^3 - 1) = 2,106\)
\(3^{x - 3}26 = 2,106\)
\(3^{x - 3} = 81\)
\(3^{x - 3} = 3^4\)
\(x-3=4\)
\(x=7\)
Answer: D.
16 Apr 2024, 10:02
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If \(3^x - 3^{x - 3} = 2,106\), what is the value of x?
Since the maximum value of x is 10, which is relatively low, we can solve this problem using a units digits pattern.
The units digit pattern for powers of 3 is the following:
3 9 7 1 3 9 7 1 3 9
So, \(3^{10}\) has a units digit of 9.
Since 2,106 has a units digit of 6, we can now check to see which units digits are 3 apart that have a difference of 6.
9 - 7 = not 6
3 - 9 = not 6
1 - 3 = not 6
7 - 1 = 6
So, counting the units digits from the left to the second 7, we find that x = 7
A. 10
B. 9
C. 8
D. 7
E. 6
Correct answer: D
Since the maximum value of x is 10, which is relatively low, we can solve this problem using a units digits pattern.
The units digit pattern for powers of 3 is the following:
3 9 7 1 3 9 7 1 3 9
So, \(3^{10}\) has a units digit of 9.
Since 2,106 has a units digit of 6, we can now check to see which units digits are 3 apart that have a difference of 6.
9 - 7 = not 6
3 - 9 = not 6
1 - 3 = not 6
7 - 1 = 6
So, counting the units digits from the left to the second 7, we find that x = 7
A. 10
B. 9
C. 8
D. 7
E. 6
Correct answer: D
