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:

\( \sqrt{\frac{49}{9}} \)
\( \frac{\sqrt{49}}{\sqrt{9}} \)
\( \frac{\sqrt{7^2}}{\sqrt{3^2}} \)
\( \frac{7}{3} \)
2\(\frac{1}{3}\)

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

7\( \sqrt{6} \) x 4\( \sqrt{4} \)
(7 x 4)\( \sqrt{6 \times 4} \)
28\( \sqrt{24} \)

Now we need to simplify the radical:

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

To add these distances, they must share the same unit so first you need to first convert the swim distance from meters (m) to kilometers (km) before adding it to the bike and run distances which are already in km. To convert 500 meters to kilometers, divide the distance by 1000 to get 0.5km then add the remaining distances:

total distance = swim + bike + run
total distance = 0.5km + 50.8km + 7.6km
total distance = 58.9km

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{-4y^5}{8y^3} \)
\( \frac{-4}{8} \) y^{(5 - 3)}
-\(\frac{1}{2}\)y²

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).