First, solve for \( \left|x - 5\right| \):

\( \left|x - 5\right| \) + 5 = -4
\( \left|x - 5\right| \) = -4 - 5
\( \left|x - 5\right| \) = -9

The value inside the absolute value brackets can be either positive or negative so (x - 5) must equal - 9 or --9 for \( \left|x - 5\right| \) to equal -9:

So, x = 14 or x = -4.

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

4\( \sqrt{8} \) + 2\( \sqrt{2} \)
4\( \sqrt{4 \times 2} \) + 2\( \sqrt{2} \)
4\( \sqrt{2^2 \times 2} \) + 2\( \sqrt{2} \)
(4)(2)\( \sqrt{2} \) + 2\( \sqrt{2} \)
8\( \sqrt{2} \) + 2\( \sqrt{2} \)

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 42 meters so the equation becomes: 2w + 2h = 42.

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

2w + 2h = 42
w + h = \( \frac{42}{2} \)
w + h = 21
w = 21 - h

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

w = 21 - 2w
3w = 21
w = \( \frac{21}{3} \)
w = 7

Since h = 2w that makes h = (2 x 7) = 14 and the area = h x w = 7 x 14 = 98 m²

A number with an exponent (b^e) consists of a base (b) raised to a power (e). The exponent indicates the number of times that the base is multiplied by itself. A base with an exponent of 1 equals the base (b¹ = b) and a base with an exponent of 0 equals 1 ( (b⁰ = 1).

To raise a term with an exponent to another exponent, retain the base and multiply the exponents: