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

9\( \sqrt{9} \) x 5\( \sqrt{7} \)
(9 x 5)\( \sqrt{9 \times 7} \)
45\( \sqrt{63} \)

Now we need to simplify the radical:

45\( \sqrt{63} \)
45\( \sqrt{7 \times 9} \)
45\( \sqrt{7 \times 3^2} \)
(45)(3)\( \sqrt{7} \)
135\( \sqrt{7} \)

32% of the water consumption is industrial use and 20% is personal use so (32% - 20%) = 12% more water is used for industrial purposes. 45,000 gallons are consumed daily so industry consumes \( \frac{12}{100} \) x 45,000 gallons = 5,400 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 52 are [1, 2, 4, 13, 26, 52]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{32}{52} \) = \( \frac{\frac{32}{4}}{\frac{52}{4}} \) = \( \frac{8}{13} \)

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{25}{49}} \)
\( \frac{\sqrt{25}}{\sqrt{49}} \)
\( \frac{\sqrt{5^2}}{\sqrt{7^2}} \)
\(\frac{5}{7}\)

The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 14 are [14, 28, 42, 56, 70, 84, 98]. The first few multiples they share are [42, 84] making 42 the smallest multiple 6 and 14 have in common.