Question

Describe the first five perfect squares (x) such that 2
`2*√(x)` square root of x is also a perfect square. Describe your method.

Answer 1

Answer: Well to find the first five perfect squares (x) such that 2 times the square root of x is also a perfect square, we can use a systematic approach.

1. Let's start by listing the perfect squares in ascending order. The first five perfect squares are 1, 4, 9, 16, and 25.

2. Next, we'll calculate 2 times the square root of each of these perfect squares. For example, for the first perfect square, 1, we have 2 times the square root of 1, which is 2. We can repeat this step for each perfect square.

- For the perfect square 1: 2 times the square root of 1 is 2.

- For the perfect square 4: 2 times the square root of 4 is 4.

- For the perfect square 9: 2 times the square root of 9 is 6.

- For the perfect square 16: 2 times the square root of 16 is 8.

- For the perfect square 25: 2 times the square root of 25 is 10.

3. Finally, we'll check if the values we obtained in step 2 are also perfect squares. In this case, all of the values we obtained (2, 4, 6, 8, and 10) are not perfect squares.

Therefore, there are no perfect squares (x) such that 2 times the square root of x is also a perfect square among the first five perfect squares (1, 4, 9, 16, and 25).

In summary, after examining the first five perfect squares, we found that none of them satisfy the condition of having 2 times the square root of x as a perfect square

Hope this helps

Answer 2

Answer: The first five perfect squares (x) such that 2 square root of x is also a perfect square are:

x = 4, because 2 square root of 4 = 2 * 2 = 4, which is a perfect square.

x = 36, because 2 square root of 36 = 2 * 6 = 12, which is a perfect square.

x = 144, because 2 square root of 144 = 2 * 12 = 24, which is a perfect square.

x = 400, because 2 square root of 400 = 2 * 20 = 40, which is a perfect square.

x = 900, because 2 square root of 900 = 2 * 30 = 60, which is a perfect square.

To find these numbers, I used the following method:

I noticed that if x is a perfect square, then it can be written as x = a^2, where a is an integer.

I also noticed that if 2 square root of x is a perfect square, then it can be written as 2 square root of x = b^2, where b is an integer.

Therefore, I can set up an equation as follows: b^2 = 2 * a

To solve for a and b, I need to find two integers that multiply to give an even number. For example, if I choose a = 2, then b = sqrt(4) = 2. If I choose a = 6, then b = sqrt(36) = 6. And so on.

I can use this method to generate as many pairs of a and b as I want, and then plug them into the equation x = a^2 to get the corresponding values of x.