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

6\( \sqrt{7} \) x 5\( \sqrt{9} \)
(6 x 5)\( \sqrt{7 \times 9} \)
30\( \sqrt{63} \)

Now we need to simplify the radical:

30\( \sqrt{63} \)
30\( \sqrt{7 \times 9} \)
30\( \sqrt{7 \times 3^2} \)
(30)(3)\( \sqrt{7} \)
90\( \sqrt{7} \)

A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

37,000 fans x \( \frac{4}{5} \) = \( \frac{148000}{5} \) = 29,600 fans.

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:

4y⁵ x 8y⁷
(4 x 8)y^{(5 + 7)}
32y¹²

The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 11 are [11, 22, 33, 44, 55, 66, 77, 88, 99]. The first few multiples they share are [33, 66, 99] making 33 the smallest multiple 3 and 11 have in common.

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{25}{9}} \)
\( \frac{\sqrt{25}}{\sqrt{9}} \)
\( \frac{\sqrt{5^2}}{\sqrt{3^2}} \)
\( \frac{5}{3} \)
1\(\frac{2}{3}\)