Question

select the correct answer from each drop-down menu.
simplify.
√63
drop - down menu √(drop - down menu)
options in first drop - down: 27, 7, 9, 3

Explanation:

Step1: Factor 63 into perfect square and another number

We know that \( 63 = 9\times7 \), where 9 is a perfect square (\( 3^2 \)). So, \( \sqrt{63}=\sqrt{9\times7} \).

Step2: Use square root property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \)

By the property of square roots, \( \sqrt{9\times7}=\sqrt{9}\times\sqrt{7} \), and \( \sqrt{9} = 3 \), but in the given format, we have \( \sqrt{9}\times\sqrt{7} \) can be written as \( 3\sqrt{7} \) or in the drop - down format, the first box should be 3 (wait, no, looking at the drop - down options: the first drop - down has 27,7,9,3. Wait, actually, \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7} \), and \( \sqrt{9} = 3 \)? No, wait, the first drop - down is the coefficient outside the square root, and the second is the number inside. Wait, \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7}=3\sqrt{7} \), but the first drop - down options are 27,7,9,3. Wait, maybe the format is \( \text{[coefficient]}\sqrt{[\text{number}]} \), so we can also write \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7} \), and if we consider the first box as the square root of the perfect square part, wait no, the first drop - down is the number outside the square root (the coefficient) and the second is the number inside. Wait, \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7} \), and \( \sqrt{9}=3 \), but 3 is an option. Wait, no, maybe I made a mistake. Wait, \( \sqrt{63}=\sqrt{9\times7} \), so we can write it as \( \sqrt{9}\times\sqrt{7} \), and \( \sqrt{9} = 3 \), but the first drop - down has 3 as an option? Wait, the first drop - down options are 27,7,9,3. Let's re - evaluate: \( 63=9\times7 \), so \( \sqrt{63}=\sqrt{9}\times\sqrt{7} \), and \( \sqrt{9} = 3 \), but also, \( \sqrt{63}=\sqrt{7\times9} \), so in the format \( \text{[x]}\sqrt{[\text{y}]} \), x is the square root of the perfect square factor, and y is the non - perfect square factor. The perfect square factor of 63 is 9 (since \( 9 = 3^2 \)), so \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7}=3\sqrt{7} \). But looking at the first drop - down options: 27,7,9,3. So the first box should be 3? No, wait, maybe the format is \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7} \), and if we write it as \( \text{[number]}\sqrt{[\text{number}]} \), the first number is \( \sqrt{9}=3 \), and the second number is 7. Wait, the first drop - down has 3 as an option, and the second has 7 as an option. Wait, let's check the options again. The first drop - down: 27,7,9,3. The second drop - down is inside the square root. So \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7} \), so we can write it as \( 3\sqrt{7} \), which is equivalent to \( \sqrt{9}\times\sqrt{7} \), but in the given format, it's \( \text{[x]}\sqrt{[\text{y}]} \), so x is 3 (since \( \sqrt{9}=3 \)) and y is 7. But wait, 9 is also an option. Wait, maybe the format is \( \sqrt{63}=\sqrt{9\times7} \), and we can write it as \( \sqrt{9}\times\sqrt{7} \), so the first box is \( \sqrt{9}=3 \)? No, the first drop - down is a number, not a square root. Wait, the first drop - down is the coefficient, so \( \sqrt{63}=3\sqrt{7} \), so the first box is 3 and the second is 7. But let's check the options: first drop - down has 3,7,9,27. Second drop - down (inside the square root) should be 7. Wait, maybe I messed up. Let's do prime factorization of 63: \( 63 = 3\times3\times7=9\times7 \). So \( \sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7}=3\sqrt{7} \). So the first drop - down (the coefficient) is 3, and the second (inside the square root) is 7. But wait, 9 is also an option. Wait, maybe the format is \( \sqrt{63}=\sqrt{9\times7} \), and we can write it as \( \sqrt{9}\times\sqrt{7} \), so if we consider the first box as the square root of 9, but no, the first box is a number. Wait, the first drop - down options are 27,7,9,3. Let's see: \( \sqrt{63}=\sqrt{9\times7} \), so if we write it as \( \sqrt{9}\times\sqrt{7} \), and \( \sqrt{9}=3 \), so the first box is 3, the second is 7.

Answer:

First drop - down: 3, Second drop - down: 7 (So in the format, it's \( 3\sqrt{7} \), so the first box is 3 and the second is 7)