If q is the square of a natural number p, then p is-

Q: If q is the square of a natural number p, then p is-

Explanation: 1. **Why Option 2 is Correct:** - If $ q $ is the square of a natural number $ p $, it means that $ q = p^2 $. To find $ p $, we need to determine the number whose square is $ q $. The mathematical operation that reverses squaring is taking the square root. Therefore, $ p $ is the square root of $ q $. This is why Option 2, "the square root of q," is the correct answer. 2. **Why the Other Options are Incorrect:** - **Option 1: "the square of q"** - This option suggests that $ p $ is $ q^2 $. However, if $ q = p^2 $, then $ p $ cannot be $ q^2 $ because that would imply $ p = (p^2)^2 $, which is not logically consistent with the definition of $ q $ as $ p^2 $. - **Option 3: "equal to q"** - This option implies that $ p = q $. However, since $ q = p^2 $, $ p $ cannot be equal to $ q $ unless $ p = 1 $, which is a specific case and not generally true for all natural numbers. Therefore, this option is incorrect. - **Option 4: "greater than q"** - This option suggests that $ p > q $. However, since $ q = p^2 $, $ p $ is actually less than or equal to $ q $ for all natural numbers $ p $ greater than 1. For example, if $ p = 2 $, then $ q = 4 $, and clearly, $ p $ is not greater than $ q $. Thus, this option is incorrect. In conclusion, the only option that logically and mathematically fits the condition $ q = p^2 $ is Option 2, where $ p $ is the square root of $ q $.

Question

Accepted Answer

Explanation:

1. **Why Option 2 is Correct:**
   - If $ q $ is the square of a natural number $ p $, it means that $ q = p^2 $. To find $ p $, we need to determine the number whose square is $ q $. The mathematical operation that reverses squaring is taking the square root. Therefore, $ p $ is the square root of $ q $. This is why Option 2, "the square root of q," is the correct answer.

2. **Why the Other Options are Incorrect:**
   - **Option 1: "the square of q"**
     - This option suggests that $ p $ is $ q^2 $. However, if $ q = p^2 $, then $ p $ cannot be $ q^2 $ because that would imply $ p = (p^2)^2 $, which is not logically consistent with the definition of $ q $ as $ p^2 $.
   
   - **Option 3: "equal to q"**
     - This option implies that $ p = q $. However, since $ q = p^2 $, $ p $ cannot be equal to $ q $ unless $ p = 1 $, which is a specific case and not generally true for all natural numbers. Therefore, this option is incorrect.
   
   - **Option 4: "greater than q"**
     - This option suggests that $ p > q $. However, since $ q = p^2 $, $ p $ is actually less than or equal to $ q $ for all natural numbers $ p $ greater than 1. For example, if $ p = 2 $, then $ q = 4 $, and clearly, $ p $ is not greater than $ q $. Thus, this option is incorrect.

In conclusion, the only option that logically and mathematically fits the condition $ q = p^2 $ is Option 2, where $ p $ is the square root of $ q $.