The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

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

2\( \sqrt{63} \) + 6\( \sqrt{7} \)
2\( \sqrt{9 \times 7} \) + 6\( \sqrt{7} \)
2\( \sqrt{3^2 \times 7} \) + 6\( \sqrt{7} \)
(2)(3)\( \sqrt{7} \) + 6\( \sqrt{7} \)
6\( \sqrt{7} \) + 6\( \sqrt{7} \)

Now that the radicands are identical, you can add them together:

The ratio of football cards to baseball cards is 5:2 and the ratio of baseball cards to basketball cards is 5:1. 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 25:10 and the ratio of baseball cards to basketball cards as 10:2. So, the ratio of football cards to basketball cards is football:baseball, baseball:basketball or 25:10, 10:2 which reduces to 25:2.

To simplify this fraction, first find the greatest common factor between them. The factors of 32 are [1, 2, 4, 8, 16, 32] and the factors of 64 are [1, 2, 4, 8, 16, 32, 64]. They share 6 factors [1, 2, 4, 8, 16, 32] making 32 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{32}{64} \) = \( \frac{\frac{32}{32}}{\frac{64}{32}} \) = \( \frac{1}{2} \)

A factor is a positive integer that divides evenly into a given number. For example, the factors of 8 are 1, 2, 4, and 8.