Problem 5 - Algebra for College Students

Question

Multiply and simplify where possible. \((4 \sqrt{2})(-6 \sqrt{5})\)

Step-by-Step Solution

Answer

The simplified product is \(-24\sqrt{10}\).

Step 1: Identify the Components

Recognize that we are multiplying two terms that both involve radicals. The first term is \(4 \sqrt{2}\) and the second term is \(-6 \sqrt{5}\).

Step 2: Multiply the Coefficients

Multiply the coefficients (numbers outside the square roots) of the two terms. This means multiplying 4 by -6. \[4 \times (-6) = -24\]

Step 3: Multiply the Radicals

Multiply the numbers inside the radicals. This means multiplying \(\sqrt{2}\) by \(\sqrt{5}\): \[\sqrt{2} \times \sqrt{5} = \sqrt{10}\]There is no need to simplify \(\sqrt{10}\) further as it does not contain a square factor.

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to get the final expression: \[-24 \times \sqrt{10} = -24\sqrt{10}\]

Key Concepts

Multiplying Radicals

When you multiply radicals, you are effectively using both the coefficients and the numbers inside the radical signs. This involves two steps. First, multiply the coefficients, which are the numbers outside the square roots, just like you would with any normal numbers. For instance, if you have \(4\sqrt{2}\) and \(-6\sqrt{5}\), you multiply the 4 and the -6. This gives you -24. Next, multiply the numbers under the radicals. Here, you multiply \(\sqrt{2}\) and \(\sqrt{5}\), resulting in \(\sqrt{10}\). After multiplying both parts, combine your results to form \(-24\sqrt{10}\). It’s crucial to remember that each part must be treated separately before combining the outcomes.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the square root sign if possible. This step comes after you have multiplied the radicals. The main goal is to check if the number under the radical can be split into factors, one of which is a perfect square. For example, for \(\sqrt{10}\), you look for factors like squares of 2, 3, or others. In this case, \(\sqrt{10}\) consists of factors only (1, 2, 5, 10), which aren't perfect squares, so \(\sqrt{10}\) remains as it is. If the number had been \(\sqrt{12}\), it would be simplified further, since \(12\) can be split into \(4 \times 3\), and 4 is a perfect square. Therefore, \(\sqrt{12}\) would simplify to \(2\sqrt{3}\). Recognizing these opportunities to simplify can help make your final expression more concise.

Coefficients

Coefficients are the numerical values multiplied by the radicals. They are essential, as they signify the magnitude or scale of the entire expression. When handling expressions like \(4\sqrt{2}\) and \(-6\sqrt{5}\), the coefficients are 4 and -6. You multiply these coefficients independently of the numbers inside the radicals. This multiplication process is straightforward: \(4 \times -6 = -24\). After you've multiplied the coefficients together, you multiply the radicals separately and then combine these results to write your final product. Coefficients affect the overall size and sign of the expression, highlighting their role in maintaining balance and accuracy in calculations.