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

9\( \sqrt{8} \) x 5\( \sqrt{6} \)
(9 x 5)\( \sqrt{8 \times 6} \)
45\( \sqrt{48} \)

Now we need to simplify the radical:

45\( \sqrt{48} \)
45\( \sqrt{3 \times 16} \)
45\( \sqrt{3 \times 4^2} \)
(45)(4)\( \sqrt{3} \)
180\( \sqrt{3} \)

To fill a 15 gallon tank exactly halfway you'll need 7\(\frac{1}{2}\) gallons of fuel. Each fuel can holds 1\(\frac{1}{2}\) gallons so:

cans = \( \frac{7\frac{1}{2} \text{ gallons}}{1\frac{1}{2} \text{ gallons}} \) = 5

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).

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{2!}{6!} \)
\( \frac{2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6 \times 5 \times 4 \times 3} \)
\( \frac{1}{360} \)

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for time:

time = \( \frac{\text{distance}}{\text{speed}} \)
time = \( \frac{45mi}{15mph} \)
3 hours