Question
Under what conditions will the expression
m
(
m
n
)
3
be a positive number?
I
.
m
<
0
and
n
>
0
II
.
m
>
0
and
n
>
0
III
.
m
<
0
and
n
<
0
IV
.
m
is any number and
n
>
0
m
(
m
n
)
3
be a positive number?
I
.
m
<
0
and
n
>
0
II
.
m
>
0
and
n
>
0
III
.
m
<
0
and
n
<
0
IV
.
m
is any number and
n
>
0
Answers
GPT 3.5
The expression \(m(mn)^3\) will be a positive number under conditions I and II only.
I. When \(m < 0\) and \(n > 0\):
If \(m < 0\), then \(m^2 > 0\). Since \(n > 0\), \((mn)^3\) will also be positive. Therefore, \(m(mn)^3\) will be positive.
II. When \(m > 0\) and \(n > 0\):
If \(m > 0\), then \(m^2 > 0\). Since \(n > 0\), \((mn)^3\) will also be positive. Therefore, \(m(mn)^3\) will be positive.
III. When \(m < 0\) and \(n < 0\):
If both \(m\) and \(n\) are negative, then \((mn)\) will be positive. However, when this positive value is raised to the power of 3, it will become negative. Therefore, \(m(mn)^3\) will not be a positive number under condition III.
IV. When \(m\) is any number and \(n > 0\):
Since \(m\) can be any number in this case, the expression \(m(mn)^3\) can be both positive and negative depending on the value of \(m\) and \(n\). Therefore, it does not necessarily result in a positive number under condition IV.
I. When \(m < 0\) and \(n > 0\):
If \(m < 0\), then \(m^2 > 0\). Since \(n > 0\), \((mn)^3\) will also be positive. Therefore, \(m(mn)^3\) will be positive.
II. When \(m > 0\) and \(n > 0\):
If \(m > 0\), then \(m^2 > 0\). Since \(n > 0\), \((mn)^3\) will also be positive. Therefore, \(m(mn)^3\) will be positive.
III. When \(m < 0\) and \(n < 0\):
If both \(m\) and \(n\) are negative, then \((mn)\) will be positive. However, when this positive value is raised to the power of 3, it will become negative. Therefore, \(m(mn)^3\) will not be a positive number under condition III.
IV. When \(m\) is any number and \(n > 0\):
Since \(m\) can be any number in this case, the expression \(m(mn)^3\) can be both positive and negative depending on the value of \(m\) and \(n\). Therefore, it does not necessarily result in a positive number under condition IV.