To multiply terms with radicals, multiply the coefficients and radicands separately:

8\( \sqrt{2} \) x 3\( \sqrt{7} \)
(8 x 3)\( \sqrt{2 \times 7} \)
24\( \sqrt{14} \)

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

To add these radicals together their radicands must be the same:

7\( \sqrt{125} \) + 8\( \sqrt{5} \)
7\( \sqrt{25 \times 5} \) + 8\( \sqrt{5} \)
7\( \sqrt{5^2 \times 5} \) + 8\( \sqrt{5} \)
(7)(5)\( \sqrt{5} \) + 8\( \sqrt{5} \)
35\( \sqrt{5} \) + 8\( \sqrt{5} \)

Now that the radicands are identical, you can add them together:

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 24 meters so the equation becomes: 2w + 2h = 24.

Putting these two equations together and solving for width (w):

2w + 2h = 24
w + h = \( \frac{24}{2} \)
w + h = 12
w = 12 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 12 - 2w
3w = 12
w = \( \frac{12}{3} \)
w = 4

Since h = 2w that makes h = (2 x 4) = 8 and the area = h x w = 4 x 8 = 32 m²

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.