To raise a term with an exponent to another exponent, retain the base and multiply the exponents:

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

4\( \sqrt{9} \) x 2\( \sqrt{5} \)
(4 x 2)\( \sqrt{9 \times 5} \)
8\( \sqrt{45} \)

Now we need to simplify the radical:

8\( \sqrt{45} \)
8\( \sqrt{5 \times 9} \)
8\( \sqrt{5 \times 3^2} \)
(8)(3)\( \sqrt{5} \)
24\( \sqrt{5} \)

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40]. The first few multiples they share are [4, 8, 12, 16, 20] making 4 the smallest multiple 2 and 4 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{5 x 2}{2 x 2} \) + \( \frac{2 x 1}{4 x 1} \)

\( \frac{10}{4} \) + \( \frac{2}{4} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{10 + 2}{4} \) = \( \frac{12}{4} \) = 3

The greatest common factor (GCF) is the greatest factor that divides two integers.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{4 x 2}{4 x 2} \) - \( \frac{5 x 1}{8 x 1} \)

\( \frac{8}{8} \) - \( \frac{5}{8} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{8 - 5}{8} \) = \( \frac{3}{8} \) = \(\frac{3}{8}\)