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:

\( \sqrt{\frac{81}{16}} \)
\( \frac{\sqrt{81}}{\sqrt{16}} \)
\( \frac{\sqrt{9^2}}{\sqrt{4^2}} \)
\( \frac{9}{4} \)
2\(\frac{1}{4}\)

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.

The factors of 52 are [1, 2, 4, 13, 26, 52] and the factors of 16 are [1, 2, 4, 8, 16]. They share 3 factors [1, 2, 4] making 4 the greatest factor 52 and 16 have in common.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{3 x 5}{9 x 5} \) + \( \frac{4 x 3}{15 x 3} \)

\( \frac{15}{45} \) + \( \frac{12}{45} \)

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

\( \frac{15 + 12}{45} \) = \( \frac{27}{45} \) = \(\frac{3}{5}\)

The factors of a number are all positive integers that divide evenly into the number. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.