60% of the water consumption is industrial use and 29% is personal use so (60% - 29%) = 31% more water is used for industrial purposes. 15,000 gallons are consumed daily so industry consumes \( \frac{31}{100} \) x 15,000 gallons = 4,650 gallons.

To simplify this fraction, first find the greatest common factor between them. The factors of 32 are [1, 2, 4, 8, 16, 32] and the factors of 44 are [1, 2, 4, 11, 22, 44]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{32}{44} \) = \( \frac{\frac{32}{4}}{\frac{44}{4}} \) = \( \frac{8}{11} \)

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{2!}{5!} \)
\( \frac{2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{5 \times 4 \times 3} \)
\( \frac{1}{60} \)

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

8\( \sqrt{8} \) x 3\( \sqrt{3} \)
(8 x 3)\( \sqrt{8 \times 3} \)
24\( \sqrt{24} \)

Now we need to simplify the radical:

24\( \sqrt{24} \)
24\( \sqrt{6 \times 4} \)
24\( \sqrt{6 \times 2^2} \)
(24)(2)\( \sqrt{6} \)
48\( \sqrt{6} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{6} \) x \( \frac{2}{8} \) = \( \frac{2 x 2}{6 x 8} \) = \( \frac{4}{48} \) = \(\frac{1}{12}\)