2019 Mock AMC 10B Problems/Problem 3

Problem 3

Which of these numbers is a rational number?

$\textbf{(A) }(\sqrt[3]{3})^{2018} \qquad \textbf{(B) }(\sqrt{3})^{2019} \qquad \textbf{(C) }(3+\sqrt{2})^2 \qquad \textbf{(D) }(2\pi)^2 \qquad \textbf{(E) }(3+\sqrt{2})(3-\sqrt{2}) \qquad$

Solution

After trying each option we have

A) $3^\frac{2018}{3}$ which is irrational as 2018 is not divisible by 3

B) $3^\frac{2019}{2}$ which is irrational as 2019 isn't divisible by 2

C) $3^2+\sqrt{2}^2+6\sqrt{2}$ which equals $11+6\sqrt{2}$ which is irrational

D) $(2\pi)^2$ equals $4\pi^2$ , which is irrational

E) $(3-\sqrt{2})(3+\sqrt{2})=9-2=7$ which is rational

Our answer is $\boxed{\textbf{(E) }(3-\sqrt{2})(3+\sqrt{2})}$ .

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2019 Mock AMC 10B Problems/Problem 3

Problem 3

Solution