A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

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:

6\( \sqrt{6} \) x 7\( \sqrt{6} \)
(6 x 7)\( \sqrt{6 \times 6} \)
42\( \sqrt{36} \)

Now we need to simplify the radical:

42\( \sqrt{36} \)
42\( \sqrt{6^2} \)
(42)(6)
252

By buying two shirts, Charlie will save $38 x \( \frac{35}{100} \) = \( \frac{$38 x 35}{100} \) = \( \frac{$1330}{100} \) = $13.30 on the second shirt.

So, his total cost will be
$38.00 + ($38.00 - $13.30)
$38.00 + $24.70
$62.70

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.