If p is a real number such that $0

Q: If p is a real number such that $0<p<1$ and x and y are real numbers with x < y, then

Explanation: 1. **Option 4: $p^{x} > p^{y}$** This option is correct. Given that $0 < p < 1$ and $x < y$, the properties of exponents and the nature of numbers between 0 and 1 dictate that raising such a number to a higher power results in a smaller value. Therefore, when $x < y$, $p^{x}$ will be greater than $p^{y}$ because $p^{x}$ involves raising $p$ to a smaller exponent compared to $p^{y}$. This is a fundamental property of numbers between 0 and 1, where increasing the exponent decreases the value. 2. **Option 1: $p^{x} p^{x} > 1$** This option is incorrect. Firstly, since $0 1 > p^{x}$** This option is incorrect. As mentioned, since $0 < p < 1$, both $p^{x}$ and $p^{y}$ will be less than 1, making it impossible for $p^{y}$ to be greater than 1. Furthermore, $p^{x}$ should be greater than $p^{y}$, not less. In conclusion, Option 4 is the only option that correctly describes the relationship between $p^{x}$ and $p^{y}$ given the conditions $0 < p < 1$ and $x < y$.

Question

If p is a real number such that $0<p<1$ and x and y are real numbers with x < y, then

Accepted Answer

Explanation: 1. **Option 4: $p^{x} > p^{y}$** This option is correct. Given that $0 < p < 1$ and $x < y$, the properties of exponents and the nature of numbers between 0 and 1 dictate that raising such a number to a higher power results in a smaller value. Therefore, when $x < y$, $p^{x}$ will be greater than $p^{y}$ because $p^{x}$ involves raising $p$ to a smaller exponent compared to $p^{y}$. This is a fundamental property of numbers between 0 and 1, where increasing the exponent decreases the value. 2. **Option 1: $p^{x} < p^{y}$** This option is incorrect. As explained above, for $0 < p < 1$, increasing the exponent results in a smaller value. Since $x < y$, $p^{x}$ should be greater than $p^{y}$, not less. Therefore, $p^{x} < p^{y}$ contradicts the behavior of exponents for numbers between 0 and 1. 3. **Option 2: $p^{y} > p^{x} > 1$** This option is incorrect. Firstly, since $0 < p < 1$, any power of $p$ will also be less than 1. Therefore, $p^{x}$ and $p^{y}$ cannot be greater than 1. Additionally, as previously explained, $p^{x}$ should be greater than $p^{y}$, not the other way around. 4. **Option 3: $p^{y} > 1 > p^{x}$** This option is incorrect. As mentioned, since $0 < p < 1$, both $p^{x}$ and $p^{y}$ will be less than 1, making it impossible for $p^{y}$ to be greater than 1. Furthermore, $p^{x}$ should be greater than $p^{y}$, not less. In conclusion, Option 4 is the only option that correctly describes the relationship between $p^{x}$ and $p^{y}$ given the conditions $0 < p < 1$ and $x < y$.