The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [40, 80] making 40 the smallest multiple 8 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{3 x 5}{8 x 5} \) + \( \frac{2 x 4}{10 x 4} \)

\( \frac{15}{40} \) + \( \frac{8}{40} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{15 + 8}{40} \) = \( \frac{23}{40} \) = \(\frac{23}{40}\)

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so subtract the coefficients and retain the base and exponent:

8b⁶ - 6b⁶
(8 - 6)b⁶
2b⁶

45% of the water consumption is industrial use and 31% is personal use so (45% - 31%) = 14% more water is used for industrial purposes. 45,000 gallons are consumed daily so industry consumes \( \frac{14}{100} \) x 45,000 gallons = 6,300 gallons.

The factors of 36 are [1, 2, 3, 4, 6, 9, 12, 18, 36] and the factors of 76 are [1, 2, 4, 19, 38, 76]. They share 3 factors [1, 2, 4] making 4 the greatest factor 36 and 76 have in common.