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Sample Question 7:
Step 1: Understand that a number is a perfect square if it can be expressed as \(x^2\) where \(x \in \mathbb{R}\).
Step 2: Divide all the representative exponents by 2.
- \(1^{2016}=(1^{1008})^{2}\)
- \(2^{2017}=2^{\frac {2017}{2}}\)
- \(3^{2018}=(3^{1009})^{2}\)
- \(4^{2019}=4^{\frac {2019}{2}}\)
- \(5^{2020}=(5^{1010})^{2}\)
Step 3: Note that 4 is a perfect square itself, so even though the exponent is odd, \(4^{2019}\) is a perfect square. This is due to \(4^{2019}=4^{2018} \cdot 4\) where the exponent \(2018\) is even and the base \(4\) is a perfect square, thus making \(4^{2019}\) a perfect square.
Step 4: With exclusion of \(4^{2019}\), choose the remaining option with the odd exponent is not a perfect square.
Step 5: Find that the only option left is \(\boxed{\textbf{(B) }2^{2017}}\).
Which of the following numbers is not a perfect square?
\(\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}\)
Answer Keys
Question 7: BSolutions
Question 7Step 1: Understand that a number is a perfect square if it can be expressed as \(x^2\) where \(x \in \mathbb{R}\).
Step 2: Divide all the representative exponents by 2.
- \(1^{2016}=(1^{1008})^{2}\)
- \(2^{2017}=2^{\frac {2017}{2}}\)
- \(3^{2018}=(3^{1009})^{2}\)
- \(4^{2019}=4^{\frac {2019}{2}}\)
- \(5^{2020}=(5^{1010})^{2}\)
Step 3: Note that 4 is a perfect square itself, so even though the exponent is odd, \(4^{2019}\) is a perfect square. This is due to \(4^{2019}=4^{2018} \cdot 4\) where the exponent \(2018\) is even and the base \(4\) is a perfect square, thus making \(4^{2019}\) a perfect square.
Step 4: With exclusion of \(4^{2019}\), choose the remaining option with the odd exponent is not a perfect square.
Step 5: Find that the only option left is \(\boxed{\textbf{(B) }2^{2017}}\).