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.

To simplify a radical, factor out the perfect squares:

\( \sqrt{112} \)
\( \sqrt{16 \times 7} \)
\( \sqrt{4^2 \times 7} \)
4\( \sqrt{7} \)

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 12 meters so the equation becomes: 2w + 2h = 12.

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

2w + 2h = 12
w + h = \( \frac{12}{2} \)
w + h = 6
w = 6 - h

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

w = 6 - 2w
3w = 6
w = \( \frac{6}{3} \)
w = 2

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

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{6 x 3}{8 x 3} \) - \( \frac{9 x 2}{12 x 2} \)

\( \frac{18}{24} \) - \( \frac{18}{24} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{18 - 18}{24} \) = \( \frac{0}{24} \) =

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

8\( \sqrt{112} \) + 9\( \sqrt{7} \)
8\( \sqrt{16 \times 7} \) + 9\( \sqrt{7} \)
8\( \sqrt{4^2 \times 7} \) + 9\( \sqrt{7} \)
(8)(4)\( \sqrt{7} \) + 9\( \sqrt{7} \)
32\( \sqrt{7} \) + 9\( \sqrt{7} \)

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