Is (r^2)x>0?
1) r^5=1
2) x>0

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For (r^2)x>0, we need to fulfill two conditions:-

either r or s must not be 0

x>0 because r^2 will always be +ve

1) r^5=1
r= 1 (as it carries odd power). But there is no information about x. Not sufficient.

2) x>0
No information about r. not sufficient.

Combining both statements fulfills our conditions. Hence sufficient.

C is the answer
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r^2 is always a positive value unless r = 0.

Statement 1 says r is not equal to zero. But it is silent on x.
x could be zero, or negative, then entire value will be zero or negative.
If x is positive, entire value will be positive

So not sufficient

Statement 2 is silent on r. Not sufficient.

When we combine, we know that x is not equal to zero and r^2 is always positive. So confirmed "Yes" can be said against the question.

C is the answer.

Target question: Is (r²)(x) > 0?

Statement 1: r⁵ = 1
This tells us that r = 1, however we don't know the value of x.
Let's TEST some values.
Case a: r = 1 and x = 1, in which case (r²)(x) = (1²)(1) = 1. Here, (r²)(x) is GREATER than 0
Case b: r = 1 and x = -1, in which case (r²)(x) = (1²)(-1) = -1. Here, (r²)(x) is NOT GREATER than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x > 0
Okay, x is POSITIVE, however we don't know the value of r.
Let's TEST some values.
Case a: r = 1 and x = 1, in which case (r²)(x) = (1²)(1) = 1. Here, (r²)(x) is GREATER than 0
Case b: r = 0 and x = 1, in which case (0²)(x) = (0²)(1) = 0. Here, (r²)(x) is NOT GREATER than 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that r = 1, which means r² = 1
Statement 2 tells us that x is POSITIVE
This means that (r²)(x) = (1³)(POSITIVE) = SOME POSITVE #. In other words, (r²)(x) is GREATER than 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer =

Cheers,
Brent
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