To add 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:

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

\( \frac{30}{15} \) + \( \frac{9}{15} \)

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

\( \frac{30 + 9}{15} \) = \( \frac{39}{15} \) = 2\(\frac{3}{5}\)

The ratio of football cards to baseball cards is 9:2 and the ratio of baseball cards to basketball cards is 9:1. To solve this problem, we need the baseball card side of each ratio to be equal so we need to rewrite the ratios in terms of a common number of baseball cards. (Think of this like finding the common denominator when adding fractions.) The ratio of football to baseball cards can also be written as 81:18 and the ratio of baseball cards to basketball cards as 18:2. So, the ratio of football cards to basketball cards is football:baseball, baseball:basketball or 81:18, 18:2 which reduces to 81:2.

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

8\( \sqrt{175} \) - 2\( \sqrt{7} \)
8\( \sqrt{25 \times 7} \) - 2\( \sqrt{7} \)
8\( \sqrt{5^2 \times 7} \) - 2\( \sqrt{7} \)
(8)(5)\( \sqrt{7} \) - 2\( \sqrt{7} \)
40\( \sqrt{7} \) - 2\( \sqrt{7} \)

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

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.

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

7\( \sqrt{7} \) x 6\( \sqrt{5} \)
(7 x 6)\( \sqrt{7 \times 5} \)
42\( \sqrt{35} \)