Hi,
Here are my two cents for this question
lets say if we write
n in term of its prime factors then
n = \(A^{P}*B^{Q}*C^{R}\), where A, B , C are prime numbers and P, Q R are non negative integers.
Now if the powers of prime factors of n are even we can say that
n is perfect square
Now Statement 1:
4
n = \(I^{2}\)
which means
2^{2}[/m]
n = \(I^{2}\)
taking square root on both the sides we have
\(\sqrt{2^{2}n}\)= \(\sqrt{I^{2}}\)
2 \(\sqrt{n}\) = I
For LHS to be equal to RHS \(\sqrt{n}\) has to be integer, This can be possible only when the powers of prime factors of
n are even) , which implies that n is perfect square
Now Statement 2
\(n^{3}\)= \(A^{3P}*B^{3Q}*C^{3R}\),
we are told that
\(n^{3}\)=\(I^{2}\)
\(A^{3P}*B^{3Q}*C^{3R}\)=\(I^{2}\)
taking square root on both sides we have
\(\sqrt{(A^{3P}*B^{3Q}*C^{3R})}\) =
Ithis means P Q R are even
this means that
n is a perfect square
Hence Answer Choice D
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