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Square roots have several important properties that make them easier to work with. One significant property is:

\(\text{If you have two square roots being multiplied, } \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\text{.}\)
This means you can combine two different square roots into a single square root of the product of the two values.

For example, if you have \(\sqrt{2} \cdot \sqrt{8}\), this simplifies to \(\sqrt{2 \cdot 8} = \sqrt{16} = 4\).

Using this property can simplify many problems involving square roots, making it much easier to solve them.

In the given exercise, we applied this property to combine \(\sqrt{\frac{7p}{6}}\) and \(\sqrt{\frac{5}{q}}\) into \(\sqrt{\frac{7p}{6} \cdot \frac{5}{q}}\), which helped us progress towards the solution.

Multiplying fractions involves simple arithmetic. Here's how you do it:

• Multiply the numerators (the top parts of the fractions) together.
• Multiply the denominators (the bottom parts of the fractions) together.

For instance, when multiplying \(\frac{2}{3}\) and \(\frac{4}{5}\):

First, multiply the numerators: \(2 \cdot 4 = 8.\)
Then, multiply the denominators: \(3 \cdot 5 = 15.\)

So, \(\frac{2}{3} \cdot \frac{4}{5} = \frac{8}{15}.\)

In our exercise, we used this method to multiply the fractions inside the square roots: \(\sqrt{\frac{7p}{6}} \cdot \sqrt{\frac{5}{q}} = \sqrt{\frac{7p \cdot 5}{6 \cdot q}} = \sqrt{\frac{35p}{6q}}\).

A rational expression is a fraction where both the numerator and the denominator are polynomials. When working with these, you often need to simplify or manipulate them in various ways.

Simplifying rational expressions can involve several steps:

• Factor both the numerator and the denominator.
• Cancel out any common factors.

Rational expressions behave similarly to regular fractions when multiplying or dividing. It is important to handle them carefully to avoid mistakes.

In the exercise, both expressions \(\frac{7p}{6}\) and \(\frac{5}{q}\) are rational expressions. When combined through multiplication inside the square root, the properties of fractions and square roots come together to simplify into one rational expression: \(\sqrt{\frac{35p}{6q}}.\)