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

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

5\( \sqrt{3} \) x 6\( \sqrt{4} \)
(5 x 6)\( \sqrt{3 \times 4} \)
30\( \sqrt{12} \)

Now we need to simplify the radical:

30\( \sqrt{12} \)
30\( \sqrt{3 \times 4} \)
30\( \sqrt{3 \times 2^2} \)
(30)(2)\( \sqrt{3} \)
60\( \sqrt{3} \)

The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [35, 70] making 35 the smallest multiple 5 and 7 have in common.

To subtract these radicals together their radicands must be the same:

8\( \sqrt{20} \) - 3\( \sqrt{5} \)
8\( \sqrt{4 \times 5} \) - 3\( \sqrt{5} \)
8\( \sqrt{2^2 \times 5} \) - 3\( \sqrt{5} \)
(8)(2)\( \sqrt{5} \) - 3\( \sqrt{5} \)
16\( \sqrt{5} \) - 3\( \sqrt{5} \)

Now that the radicands are identical, you can subtract them: