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:

\( \sqrt{\frac{4}{64}} \)
\( \frac{\sqrt{4}}{\sqrt{64}} \)
\( \frac{\sqrt{2^2}}{\sqrt{8^2}} \)
\(\frac{1}{4}\)

1

To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]. They share 8 factors [1, 2, 4, 5, 8, 10, 20, 40] making 40 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{40}{80} \) = \( \frac{\frac{40}{40}}{\frac{80}{40}} \) = \( \frac{1}{2} \)

The greatest common factor (GCF) is the greatest factor that divides two integers.

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 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

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

\( \frac{9 x 3}{8 x 3} \) + \( \frac{3 x 2}{12 x 2} \)

\( \frac{27}{24} \) + \( \frac{6}{24} \)

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

\( \frac{27 + 6}{24} \) = \( \frac{33}{24} \) = 1\(\frac{3}{8}\)