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# Roots: Perfect squares

Are both $$x$$ and $$y$$ integers? >(1) $$\sqrt{x}+\sqrt{y}$$ is an integer. >(2) $$\frac{x}{y}$$ is an integer.
Correct. [[snippet]] Stat. (1): You can plug in integer values for $$x$$ and $$y$$, such as $$x=y=4$$, so that >$$\sqrt{x}+\sqrt{y}= 2+2 = 4$$, yielding an integer result and an answer of "Yes." However, plugging in $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ is also an option since $$\frac{1}{2}+\frac{1}{2} = 1$$ is an integer, but $$x$$ and $$y$$ are not integers, yielding an answer of "No." There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. Stat. (2): Try to use the same Plug-Ins you chose for Stat. (1): if $$x=y=4$$, then >$$\frac{x}{y}=\frac{4}{4}=1$$. The number 1 is an integer, so these values are possible, and the answer is "Yes." However, $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ also satisfies Stat. (2), since $$x$$ and $$y$$ are the same number, and $$\frac{x}{y}=1$$. Yet $$x$$ and $$y$$ are not integers, so this yields an answer of "No." There is no definite answer, so **Stat.(2) → Maybe → IS → CE**. Stat. (1+2): Since the same sets of numbers of $$x=y=4$$ and $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ satisfy both statements and still reach an answer of "Yes" and "No," combining the statements is still insufficient. **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] Stat. (1): You can plug in integer values for $$x$$ and $$y$$, such as $$x=y=4$$, so that >$$\sqrt{x}+\sqrt{y}= 2+2 = 4$$, yielding an integer result and an answer of "Yes." However, plugging in $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ is also an option since $$\frac{1}{2}+\frac{1}{2} = 1$$ is an integer, but $$x$$ and $$y$$ are not integers, yielding an answer of "No." There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] Stat. (1+2): Since the same sets of numbers of $$x=y=4$$ and $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ satisfy both statements and still reach an answer of "Yes" and "No," combining the statements is still insufficient. **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] Stat. (1): You can plug in integer values for $$x$$ and $$y$$, such as $$x=y=4$$, so that >$$\sqrt{x}+\sqrt{y}= 2+2 = 4$$, yielding an integer result and an answer of "Yes." However, plugging in $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ is also an option since $$\frac{1}{2}+\frac{1}{2} = 1$$ is an integer, but $$x$$ and $$y$$ are not integers, yielding an answer of "No." There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. Stat. (2): Try to use the same Plug-Ins you chose for Stat. (1): if $$x=y=4$$, then >$$\frac{x}{y}=\frac{4}{4}=1$$. The number 1 is an integer, so these values are possible, and the answer is "Yes." However, $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ also satisfies Stat. (2), since $$x$$ and $$y$$ are the same number, and $$\frac{x}{y}=1$$. Yet $$x$$ and $$y$$ are not integers, so this yields an answer of "No." There is no definite answer, so **Stat.(2) → Maybe → IS → CE**.
Incorrect. [[snippet]] Stat. (2): Try to use the same Plug-Ins you chose for Stat. (1): if $$x=y=4$$, then >$$\frac{x}{y}=\frac{4}{4}=1$$. The number 1 is an integer, so these values are possible, and the answer is "Yes." However, $$\sqrt{x}=\sqrt{y}=\frac{1}{2}$$ also satisfies Stat. (2), since $$x$$ and $$y$$ are the same number, and $$\frac{x}{y}=1$$. Yet $$x$$ and $$y$$ are not integers, so this yields an answer of "No." There is no definite answer, so **Stat.(2) → Maybe → IS → ACE**.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.