The number $x$ is expressed as $x={{2}^{3}}\times {{3}^{2}}\times {{y}^{3}}\times {{z}^{2}}$, wh

Question

The number $x$ is expressed as $x={{2}^{3}}\times {{3}^{2}}\times {{y}^{3}}\times {{z}^{2}}$, where $y$ and $z$ are prime numbers larger than $3$. Mark claimed that $x$ is a perfect square while Henry claimed that $x$ is a perfect cube. Determine who is correct. Provide reason(s) to support your answer.

Accepted Answer

The number $x$ is expressed as $x={{2}^{3}}\times {{3}^{2}}\times {{y}^{3}}\times {{z}^{2}}$, where $y$ and $z$ are prime numbers larger than $3$. Mark claimed that $x$ is a perfect square while Henry claimed that $x$ is a perfect cube. Determine who is correct. Provide reason(s) to support your answer. A perfect square has all even powers in its prime factorization. Since, $y$ and $z$ are both prime numbers, the prime factorization of $x$ is as expressed $2{}^\text{3}\times 3{}^\text{2}\times y{}^\text{3}\times z{}^\text{2}$ Since not all powers are even, $x$ is not a perfect square. A perfect cube has all multiples of $3$ as powers in its prime factorization. Since not all powers are divisible by $3$, $x$ is not a perfect cube. Thus, x is neither a perfect square or a perfect cube. Conclusion, neither Mark nor Henry is correct.

Perfect Square vs Perfect Cube

Perfect Square vs Perfect Cube

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