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Understanding the properties of square roots is essential when working with expressions containing square roots. One key property is that the product of two square roots can be simplified into a single square root.

More simply, the property can be stated as \(\sqrt{a} \sqrt{b} = \sqrt{a \cdot b}\).

This property helps in reducing complex expressions into simpler ones. For this specific exercise, it allows us to transform \(\sqrt{\frac{2}{t}} \sqrt{\frac{3 s}{11}}\) into \(\sqrt{\frac{2}{t} \cdot \frac{3 s}{11}}\).

Combining expressions under a single square root makes it easier to handle them in subsequent steps. Keep this property in mind whenever you encounter square roots in your mathematical problems!

When multiplying fractions, you multiply the numerators together and the denominators together. This rule helps to simplify expressions involving fractions.

For example, consider the multiplication in this exercise: \(\frac{2}{t} \cdot \frac{3 s}{11}\).

Multiplying the numerators (2 and 3s) and the denominators (t and 11) separately, we get:

• Numerator: 2 \cdot 3s = 6s
• Denominator: t \cdot 11 = 11t

Therefore, \(\frac{2}{t} \cdot \frac{3 s}{11} = \frac{6 s}{11 t}\).

Remember, keeping track of the individual parts during multiplication and simplifying them correctly is crucial.

Simplifying expressions is about reducing them to their simplest form. It makes complex calculations easier and more understandable.

In our example, once we've multiplied the fractions, we simplify what remains under the square root as follows:

• Originally, \(\sqrt{\frac{2}{t}} \cdot \sqrt{\frac{3 s}{11}}\).
• Using the square roots property, it became \(\sqrt{\frac{2}{t} \cdot \frac{3 s}{11}}\).
• Multiplying the fractions, we get \(\sqrt{\frac{6 s}{11 t}}\).

The expression is now simplified under a single square root, making it easier to understand and work with.

This process of combining and reducing terms is what makes mathematical problems more manageable.