To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

4b⁵ x 8b⁶
(4 x 8)b^{(5 + 6)}
32b¹¹

To divide fractions, invert the second fraction and then multiply:

\( \frac{2}{6} \) ÷ \( \frac{3}{9} \) = \( \frac{2}{6} \) x \( \frac{9}{3} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{6} \) x \( \frac{9}{3} \) = \( \frac{2 x 9}{6 x 3} \) = \( \frac{18}{18} \) = 1

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

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

\( \frac{8 x 3}{8 x 3} \) + \( \frac{6 x 2}{12 x 2} \)

\( \frac{24}{24} \) + \( \frac{12}{24} \)

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

\( \frac{24 + 12}{24} \) = \( \frac{36}{24} \) = 1\(\frac{1}{2}\)

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

6\( \sqrt{7} \) x 5\( \sqrt{2} \)
(6 x 5)\( \sqrt{7 \times 2} \)
30\( \sqrt{14} \)

The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [21, 42, 63, 84] making 21 the smallest multiple 3 and 7 have in common.