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 48 meters so the equation becomes: 2w + 2h = 48.

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

2w + 2h = 48
w + h = \( \frac{48}{2} \)
w + h = 24
w = 24 - h

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

w = 24 - 2w
3w = 24
w = \( \frac{24}{3} \)
w = 8

Since h = 2w that makes h = (2 x 8) = 16 and the area = h x w = 8 x 16 = 128 m²

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

5\( \sqrt{18} \) + 2\( \sqrt{2} \)
5\( \sqrt{9 \times 2} \) + 2\( \sqrt{2} \)
5\( \sqrt{3^2 \times 2} \) + 2\( \sqrt{2} \)
(5)(3)\( \sqrt{2} \) + 2\( \sqrt{2} \)
15\( \sqrt{2} \) + 2\( \sqrt{2} \)

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

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so subtract the coefficients and retain the base and exponent:

9x⁷ - 4x⁷
(9 - 4)x⁷
5x⁷

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

8\( \sqrt{63} \) - 3\( \sqrt{7} \)
8\( \sqrt{9 \times 7} \) - 3\( \sqrt{7} \)
8\( \sqrt{3^2 \times 7} \) - 3\( \sqrt{7} \)
(8)(3)\( \sqrt{7} \) - 3\( \sqrt{7} \)
24\( \sqrt{7} \) - 3\( \sqrt{7} \)

Now that the radicands are identical, you can subtract them:

The number of students taking German or Spanish is 15 + 9 = 24. Of that group of 24, 4 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 24 - 4 = 20 who are taking at least one language. 27 - 20 = 7 students who are not taking either language.