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

5\( \sqrt{112} \) - 7\( \sqrt{7} \)
5\( \sqrt{16 \times 7} \) - 7\( \sqrt{7} \)
5\( \sqrt{4^2 \times 7} \) - 7\( \sqrt{7} \)
(5)(4)\( \sqrt{7} \) - 7\( \sqrt{7} \)
20\( \sqrt{7} \) - 7\( \sqrt{7} \)

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

The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 6 and 12 have in common.

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{8} \) x \( \frac{2}{7} \) = \( \frac{2 x 2}{8 x 7} \) = \( \frac{4}{56} \) = \(\frac{1}{14}\)

The number of students taking German or Spanish is 12 + 6 = 18. Of that group of 18, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 18 - 5 = 13 who are taking at least one language. 18 - 13 = 5 students who are not taking either language.