Problem 9 - Intermediate Algebra

Question

For Problems \(1-14\), multiply and simplify where possible. $$ (5 \sqrt{6})(4 \sqrt{6}) $$

Step-by-Step Solution

Answer

The product is 120.

Step 1: Multiply the Coefficients

First, identify and multiply the coefficients, which are the numbers in front of the square roots. Here, multiply 5 and 4: \[ 5 \times 4 = 20 \]

Step 2: Multiply the Square Roots

Next, multiply the square roots. Since both are \( \sqrt{6} \), this is: \[ \sqrt{6} \times \sqrt{6} = \sqrt{6^2} = \sqrt{36} = 6 \]

Step 3: Multiply the Results of Steps 1 and 2

Now, multiply the results from Step 1 and Step 2: \[ 20 \times 6 = 120 \]

Step 4: Simplify the Expression

The expression is already simplified after performing the above multiplications. The final simplified product is 120.

Key Concepts

Simplifying Expressions

When you're confronted with an expression that involves multiplying radicals, it might initially appear complex, but it doesn't have to be. To simplify such expressions, we should follow a systematic approach. The first step is to identify like terms and use the properties of multiplication to simplify them.
In the example\((5 \sqrt{6})(4 \sqrt{6})\), we notice that the expression consists of two parts: coefficients and square roots. Our task is to break the operation into more manageable pieces. By separating these into two groups, coefficients and square roots, we can deal with each separately before combining them for a final answer.
Another crucial part of simplifying expressions is recognizing when terms are already simplified and ensuring we're not overcomplicating the problem. After multiplying and simplifying each part of the expression, confirm that nothing further can be reduced. This leads us directly to our final simplified result, which in our case is 120. This result means all terms have been multiplied and reduced to their simplest form.

Coefficients

Coefficients are the numerical parts of terms that appear in front of square roots or other variables. They are crucial when simplifying expressions because they tell us how many of a particular term we have.
In our problem, the coefficients are 5 and 4, which are the numbers directly in front of each radical. The first step in simplifying the expression is to multiply these coefficients together:
\[5 \times 4 = 20\]. This operation is carried out the same way we would multiply any other numbers, focusing only on the coefficients first without worrying about the radicals they accompany.
Multiplying coefficients helps streamline the expression, paving the way for managing the radicals separately. Remember, coefficients should always be handled before moving on to the radicals to ensure clear and smooth calculation.

Square Roots

Square roots can seem daunting, but when you break down the multiplication of square roots, they follow straightforward rules. In the expression \((5 \sqrt{6})(4 \sqrt{6})\), we encounter \(\sqrt{6}\) being multiplied by itself.
To simplify this, recall that:

• Multiplying a square root by itself results in the number inside the root, i.e., \(\sqrt{a} \times \sqrt{a} = a\).
• In our scenario, \(\sqrt{6} \times \sqrt{6}\) transforms into \(\sqrt{6^2}\) which simplifies to \(\sqrt{36}\), and \(\sqrt{36} = 6\).

After handling the square roots as described, combine the result with the product of the coefficients already calculated. Multiplying these components gives us the complete simplified expression. Understanding these basic properties of square roots simplifies solving expressions involving them considerably.