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# Roots: Overview

If $$p$$ and $$q$$ are two nonzero integers, is $$p^q \gt q^p$$? >(1) $$p = \sqrt{q}$$ >(2) $$p \ne 1$$
Incorrect. [[snippet]] According to Stat. (1), $$p = \sqrt{q}$$. Plug in values for the variables: * If $$p = 1$$ and $$q = 1$$, then $$p^q = 1^1$$ is equal to $$q^p = 1^1$$, so the answer is "No." * If $$p = 2$$ and $$q = 4$$, then $$p^q = 2^4$$ is equal to $$q^p = 4^2$$, so the answer is still "No." * If $$p = 3$$ and $$q = 9$$, then $$p^q = 3^9$$ is greater than $$q^p = 9^3$$, so the answer is finally "Yes." Thus, there is no definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] According to Stat. (2), $$p \ne 1$$. Plug in values for the variables: * If $$p = 2$$ and $$q = 4$$, then $$p^q = 2^4$$ is equal to $$q^p = 4^2$$, so the answer is "No." * If $$p = 3$$ and $$q = 9$$, then $$p^q = 3^9$$ is greater than $$q^p = 9^3$$, so the answer is "Yes." Thus, there is no definite answer, so **Stat.(2) → Maybe → IS → ACE**.
Incorrect. [[snippet]] According to Stat. (1+2), you can plug in values for the variables: * If $$p = 2$$ and $$q = 4$$, then $$p^q = 2^4$$ is equal to $$q^p = 4^2$$, so the answer is "No." * If $$p = 3$$ and $$q = 9$$, then $$p^q = 3^9$$ is greater than $$q^p = 9^3$$, so the answer is "Yes." Thus, there is still no definite answer, so **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] According to Stat. (1), $$p = \sqrt{q}$$. Plug in values for the variables: * If $$p = 1$$ and $$q = 1$$, then $$p^q = 1^1$$ is equal to $$q^p = 1^1$$, so the answer is "No." * If $$p = 2$$ and $$q = 4$$, then $$p^q = 2^4$$ is equal to $$q^p = 4^2$$, so the answer is still "No." * If $$p = 3$$ and $$q = 9$$, then $$p^q = 3^9$$ is greater than $$q^p = 9^3$$, so the answer is finally "Yes." Thus, there is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$p \ne 1$$. Although you cannot use $$p=q=1$$, you can still use the second plug-in, $$p = 2$$ and $$q = 4$$, and the third plug-in, $$p = 3$$ and $$q = 9$$. Thus, there is no definite answer, so **Stat.(2) → Maybe → IS → CE**.
Correct. [[snippet]] According to Stat. (1), $$p = \sqrt{q}$$. Plug in values for the variables: * If $$p = 1$$ and $$q = 1$$, then $$p^q = 1^1$$ is equal to $$q^p = 1^1$$, so the answer is "No." * If $$p = 2$$ and $$q = 4$$, then $$p^q = 2^4$$ is equal to $$q^p = 4^2$$, so the answer is still "No." * If $$p = 3$$ and $$q = 9$$, then $$p^q = 3^9$$ is greater than $$q^p = 9^3$$, so the answer is finally "Yes." Thus, there is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$p \ne 1$$. Although you cannot use $$p=q=1$$, you can still use the second plug-in, $$p = 2$$ and $$q = 4$$, and the third plug-in, $$p = 3$$ and $$q = 9$$. Thus, there is no definite answer, so **Stat.(2) → Maybe → IS → CE**. According to Stat. (1+2), you can use the same plug-ins as you used for Stat. (2). Thus, there is still no definite answer, so **Stat.(1+2) → Maybe → IS → E**.
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.