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

4\( \sqrt{9} \) x 5\( \sqrt{5} \)
(4 x 5)\( \sqrt{9 \times 5} \)
20\( \sqrt{45} \)

Now we need to simplify the radical:

20\( \sqrt{45} \)
20\( \sqrt{5 \times 9} \)
20\( \sqrt{5 \times 3^2} \)
(20)(3)\( \sqrt{5} \)
60\( \sqrt{5} \)

1

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50]. The first few multiples they share are [15, 30, 45, 60, 75] making 15 the smallest multiple 3 and 5 share.

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

\( \frac{5 x 5}{3 x 5} \) - \( \frac{6 x 3}{5 x 3} \)

\( \frac{25}{15} \) - \( \frac{18}{15} \)

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

\( \frac{25 - 18}{15} \) = \( \frac{7}{15} \) = \(\frac{7}{15}\)

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, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.