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The Product Rule for Radicals is a helpful tool when dealing with multiplication of square roots. It states that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). This means when multiplying two radicals, you can multiply the numbers inside the square roots together and place them under a single square root.

In our example, we have \( (4 \sqrt{3})(2 \sqrt{15}) \). Using the product rule, we combine the radicals as follows: the product of \( \sqrt{3} \) and \( \sqrt{15} \) becomes \( \sqrt{3 \times 15} \).

• Multiply 4 and 2 outside the square root, giving us \( 8 \).
• Multiply the numbers inside the square root, obtaining \( \sqrt{45} \).

This rule simplifies the process, reducing pairs of radical expressions to a singular form which can then be simplified further.

Simplifying radicals involves rewriting a square root expression in its simplest form. This means reducing the value under the square root sign into its smallest possible components.

In our problem, we saw the expression \( 8 \sqrt{45} \). To simplify \( \sqrt{45} \), first break down 45 into its factors. Notice that 45 is the product of 9 and 5, where 9 is a perfect square.

After recognizing 9 as a perfect square, rewrite \( \sqrt{45} \) as \( \sqrt{9 \times 5} \). From here, you can take the square root of 9, which is 3. Thus, \( \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \).

This reduces our entire expression to \( 24\sqrt{5} \), simplifying not only the radical but also the whole expression.

Perfect squares are numbers that are the square of an integer. Recognizing perfect squares is crucial when simplifying radical expressions.

For instance, in our example, the number 9 is a perfect square as it equals \( 3^2 \). Spotting such numbers within a radicand allows us to simplify square roots efficiently.

• Common perfect squares include numbers like 1, 4, 9, 16, 25, etc.
• Factoring out perfect squares helps decompose complex radicals into simpler parts.

By rewriting \( \sqrt{45} \) as \( \sqrt{9 \times 5} \), and recognizing that \( \sqrt{9} \) simplifies directly to 3, we drastically simplify the expression. Being able to identify and extract perfect squares is key to mastering radical simplification.