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If m and n are positive numbers, and m^5 = 200 - n^5, what is the
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12 Nov 2021, 06:47
12 Nov 2021, 06:47
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If \(m\) and \(n\) are positive numbers, and \(m^5 = 200 - n^5\), what is the greatest possible value of \(n\) ?
(A) Between 1 and 2
(B) Between 2 and 3
(C) Between 3 and 4
(D) Between 4 and 5
(E) Between 5 and 6
(A) Between 1 and 2
(B) Between 2 and 3
(C) Between 3 and 4
(D) Between 4 and 5
(E) Between 5 and 6
Retired Moderator
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Re: If m and n are positive numbers, and m^5 = 200 - n^5, what is the
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12 Nov 2021, 07:49
12 Nov 2021, 07:49
1
Start here: we know that m^5 is a positive number.
Therefore, 200-n^5 must also be greater than zero. That implies that n^5 must be LESS THAN 200.
2^5 = 32 and 3^5 = 243. 200 is of course somewhere in this range.
This means that the maximum value for n would be somewhere between 2 and 3.
(Note that m and n can be any number, not just integers because the question says "positive numbers" rather than "positive integers." Pay attention to wording in cases such as this).
Therefore, 200-n^5 must also be greater than zero. That implies that n^5 must be LESS THAN 200.
2^5 = 32 and 3^5 = 243. 200 is of course somewhere in this range.
This means that the maximum value for n would be somewhere between 2 and 3.
(Note that m and n can be any number, not just integers because the question says "positive numbers" rather than "positive integers." Pay attention to wording in cases such as this).
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Posts: 6218
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Re: If m and n are positive numbers, and m^5 = 200 - n^5, what is the
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12 Nov 2021, 08:07
12 Nov 2021, 08:07
1
Carcass wrote:
If \(m\) and \(n\) are positive numbers, and \(m^5 = 200 - n^5\), what is the greatest possible value of \(n\) ?
(A) Between 1 and 2
(B) Between 2 and 3
(C) Between 3 and 4
(D) Between 4 and 5
(E) Between 5 and 6
(A) Between 1 and 2
(B) Between 2 and 3
(C) Between 3 and 4
(D) Between 4 and 5
(E) Between 5 and 6
Given: \(m^5 = 200 - n^5\)
Add \(n^5\) to both sides of the equation to get: \(m^5 + n^5 = 200 \)
We can MAXIMIZE the value of \(n\) by MINIMIZING the value of \(m\).
We're told that \(m\) is positive, so the smallest possible value of \(m\) will be just a tiny bit bigger than 0.
Since there is very little difference between a number like \(0.00000000001\) and \(0\), let's just make things easy on ourselves and let \(m = 0\)
When we do this we get: \(0^5 + n^5 = 200\)
Simplify: \(n^5 = 200\)
We know that \(2^5 = 32\) and \(3^5 = 243\)
Since \(200\) is between \(32\) and \(243\), we know that the value of \(n\) must be between \(2\) and \(3\)
Answer: B
