A rectangular window is 4x inches wide and 8x inches long. What is an expression for the distance from one corner of the window to the opposite corner? Give your answer as a simplified radical.

The sides are 4x*1 and 4x*2, the diagonal is

4x√(1^2 + 2^2) = 4x√5

To find the distance from one corner of the window to the opposite corner, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the width of the window is 4x inches, and the length is 8x inches. Let's label the width as side a and the length as side b.

Using the Pythagorean theorem, we have:

hypotenuse^2 = a^2 + b^2

Let's substitute the values:

hypotenuse^2 = (4x)^2 + (8x)^2

Simplifying:

hypotenuse^2 = 16x^2 + 64x^2

hypotenuse^2 = 80x^2

To find the length (or the hypotenuse), we take the square root of both sides:

hypotenuse = sqrt(80x^2)

Simplifying further:

hypotenuse = sqrt(16 * 5 * x^2)

Taking out the perfect square factor:

hypotenuse = sqrt(16) * sqrt(5) * sqrt(x^2)

Simplifying the perfect square root:

hypotenuse = 4 * sqrt(5) * x

Therefore, the expression for the distance from one corner of the window to the opposite corner is 4sqrt(5)x inches.

To find the distance from one corner of the window to the opposite corner, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the two sides of the right triangle are the width and length of the window. Let's call the distance from one corner to the opposite corner "d".

Applying the Pythagorean theorem, we have:

d^2 = (4x)^2 + (8x)^2

Expanding the squares, we get:

d^2 = 16x^2 + 64x^2

Combining like terms, we have:

d^2 = 80x^2

To simplify the expression, we can take the square root of both sides:

d = √(80x^2)

Simplifying further, we can break down the radicand into its prime factors:

d = √(2^4 * 5 * x^2)

Using properties of radicals, we can separate the factors under the square root:

d = 2^2 * √(5) * √(x^2)

Simplifying, we get the final expression:

d = 4x√5