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Simplifying radicals is the process of rewriting a radical expression in its most reduced or simplest form. Sometimes, a number inside the square root sign can be broken down into smaller parts that have perfect squares. But what does that mean?
For simplicity, think about breaking down numbers into their factors. When a number inside a square root can be factored into a pair of numbers where one is a perfect square (like 4, 9, or 16), it can be simplified further. For instance, \[ \ \sqrt{18} = \sqrt{9 \times 2} \] Becomes \[ \ \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \] Because the square root of 9 is 3.

• Identify any perfect square factors inside the radical.
• Simplify by moving those perfect squares outside of the radical.

Don't forget, if no perfect square exists, like in the case of \(\sqrt{10}\), there’s nothing to simplify!

Multiplying square roots might seem tricky, but it doesn't have to be! Whenever you have two square root expressions to multiply, there’s a handy property that can make your task easier. It states: \[ \ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \] This property allows us to combine the numbers under a single square root sign. It's like working a math shortcut!
In our original exercise:

• Take \(\sqrt{2}\) and \(\sqrt{5}\).
• Use the square root multiplication property \(\sqrt{2 \times 5} = \sqrt{10}\).

So in summary, combine the square root numbers and then simplify if possible. In some cases, the new number might be simplified further. But if not, like \(\sqrt{10}\), you're already done!

Radicals, or expressions with roots like square roots, have several important properties that make simplifying and working with them more manageable. Let’s look at some crucial ones:

• Product Property: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). This lets us combine roots in multiplication.
• Quotient Property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) if \(b eq 0\). This helps when dividing expressions with roots.
• Simplifying Radicals: Identify and pull out the perfect square factors from under the radical.

For multiplication, especially, the product property is most relevant. It's the trick we use to greatly simplify expressions and ease our calculations. Remember these properties, and you'll unlock the full potential of radicals!