To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40]. The first few multiples they share are [4, 8, 12, 16, 20] making 4 the smallest multiple 2 and 4 share.

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

\( \frac{3 x 2}{2 x 2} \) + \( \frac{6 x 1}{4 x 1} \)

\( \frac{6}{4} \) + \( \frac{6}{4} \)

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

\( \frac{6 + 6}{4} \) = \( \frac{12}{4} \) = 3

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

6\( \sqrt{18} \) + 7\( \sqrt{2} \)
6\( \sqrt{9 \times 2} \) + 7\( \sqrt{2} \)
6\( \sqrt{3^2 \times 2} \) + 7\( \sqrt{2} \)
(6)(3)\( \sqrt{2} \) + 7\( \sqrt{2} \)
18\( \sqrt{2} \) + 7\( \sqrt{2} \)

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 54 meters so the equation becomes: 2w + 2h = 54.

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

2w + 2h = 54
w + h = \( \frac{54}{2} \)
w + h = 27
w = 27 - h

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

w = 27 - 2w
3w = 27
w = \( \frac{27}{3} \)
w = 9

Since h = 2w that makes h = (2 x 9) = 18 and the area = h x w = 9 x 18 = 162 m²

The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

47% of the water consumption is industrial use and 34% is personal use so (47% - 34%) = 13% more water is used for industrial purposes. 25,000 gallons are consumed daily so industry consumes \( \frac{13}{100} \) x 25,000 gallons = 3,250 gallons.