To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

5z⁵ x 6z⁴
(5 x 6)z^{(5 + 4)}
30z⁹

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

34% of the water consumption is industrial use and 10% is personal use so (34% - 10%) = 24% more water is used for industrial purposes. 10,000 gallons are consumed daily so industry consumes \( \frac{24}{100} \) x 10,000 gallons = 2,400 gallons.

The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

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