A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

35% of the water consumption is industrial use and 12% is personal use so (35% - 12%) = 23% more water is used for industrial purposes. 5,000 gallons are consumed daily so industry consumes \( \frac{23}{100} \) x 5,000 gallons = 1,150 gallons.

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

\( \frac{-x^9}{x^2} \)
\( \frac{-1}{1} \) x^{(9 - 2)}
-x⁷

To multiply terms with radicals, multiply the coefficients and radicands separately:

2\( \sqrt{5} \) x 5\( \sqrt{8} \)
(2 x 5)\( \sqrt{5 \times 8} \)
10\( \sqrt{40} \)

Now we need to simplify the radical:

10\( \sqrt{40} \)
10\( \sqrt{10 \times 4} \)
10\( \sqrt{10 \times 2^2} \)
(10)(2)\( \sqrt{10} \)
20\( \sqrt{10} \)

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

9\( \sqrt{125} \) - 7\( \sqrt{5} \)
9\( \sqrt{25 \times 5} \) - 7\( \sqrt{5} \)
9\( \sqrt{5^2 \times 5} \) - 7\( \sqrt{5} \)
(9)(5)\( \sqrt{5} \) - 7\( \sqrt{5} \)
45\( \sqrt{5} \) - 7\( \sqrt{5} \)

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