To simplify a radical, factor out the perfect squares:

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

To fill a 9 gallon tank exactly halfway you'll need 4\(\frac{1}{2}\) gallons of fuel. Each fuel can holds 1\(\frac{1}{2}\) gallons so:

cans = \( \frac{4\frac{1}{2} \text{ gallons}}{1\frac{1}{2} \text{ gallons}} \) = 3

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

5\( \sqrt{2} \) x 6\( \sqrt{4} \)
(5 x 6)\( \sqrt{2 \times 4} \)
30\( \sqrt{8} \)

Now we need to simplify the radical:

30\( \sqrt{8} \)
30\( \sqrt{2 \times 4} \)
30\( \sqrt{2 \times 2^2} \)
(30)(2)\( \sqrt{2} \)
60\( \sqrt{2} \)

The factors of 56 are [1, 2, 4, 7, 8, 14, 28, 56] and the factors of 56 are [1, 2, 4, 7, 8, 14, 28, 56]. They share 8 factors [1, 2, 4, 7, 8, 14, 28, 56] making 56 the greatest factor 56 and 56 have in common.

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

2\( \sqrt{75} \) - 2\( \sqrt{3} \)
2\( \sqrt{25 \times 3} \) - 2\( \sqrt{3} \)
2\( \sqrt{5^2 \times 3} \) - 2\( \sqrt{3} \)
(2)(5)\( \sqrt{3} \) - 2\( \sqrt{3} \)
10\( \sqrt{3} \) - 2\( \sqrt{3} \)

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