Problem 20 - Intermediate Algebra

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ 4 \sqrt{2}(3 \sqrt{12}+7 \sqrt{6}) $$

Step-by-Step Solution

Answer

The product in simplest form is \(24\sqrt{6} + 56\sqrt{3}\).

Step 1: Distribute the Expression

To simplify the expression \(4 \sqrt{2}(3 \sqrt{12} + 7 \sqrt{6})\), we need to distribute \(4 \sqrt{2}\) across each term inside the parentheses. This step involves multiplying \(4 \sqrt{2}\) by \(3 \sqrt{12}\) and \(4 \sqrt{2}\) by \(7 \sqrt{6}\) individually.

Step 2: Multiply the First Terms

Multiply \(4 \sqrt{2}\) by \(3 \sqrt{12}\).\[4 \sqrt{2} \times 3 \sqrt{12} = 12 \sqrt{24}\]Note that we multiplied the coefficients (\(4\) and \(3\)) to get \(12\), and multiplied the radicals (\(\sqrt{2}\) and \(\sqrt{12}\)) to get \(\sqrt{24}\).

Step 3: Simplify \(\sqrt{24}\)

Simplify \(\sqrt{24}\) as follows:\[\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \sqrt{6}\]Therefore, \(12 \sqrt{24} = 12 \times 2 \sqrt{6} = 24 \sqrt{6}\).

Step 4: Multiply the Second Terms

Multiply \(4 \sqrt{2}\) by \(7 \sqrt{6}\).\[4 \sqrt{2} \times 7 \sqrt{6} = 28 \sqrt{12}\]This involves multiplying the coefficients (\(4\) and \(7\)) to get \(28\), and the radicals (\(\sqrt{2}\) and \(\sqrt{6}\)) to get \(\sqrt{12}\).

Step 5: Simplify \(\sqrt{12}\)

Simplify \(\sqrt{12}\) as follows:\[\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}\]Therefore, \(28 \sqrt{12} = 28 \times 2 \sqrt{3} = 56 \sqrt{3}\).

Step 6: Combine Like Terms

Now, combine the simplified terms from Steps 3 and 5:\[24 \sqrt{6} + 56 \sqrt{3}\]These two terms are different radicals and cannot be combined further, so this is our simplified result.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that makes dealing with expressions more straightforward. It says that when you multiply a number by a sum, you can distribute the multiplier to each term inside the parentheses. For example, in the given expression

• \(4 \sqrt{2}(3 \sqrt{12} + 7 \sqrt{6})\)

we use the distributive property by multiplying \(4 \sqrt{2}\) by both \(3 \sqrt{12}\) and \(7 \sqrt{6}\). This helps simplify the expression step by step.
Make sure to multiply both the coefficients (the numbers outside the radicals) and the radicals themselves.
Using the distributive property can make expressions that look complicated more manageable and helps ensure you're applying operations correctly.

Multiplying Radicals

Multiplying radicals involves a few simple steps. When multiplying numbers under radical signs, known as radicands, you can multiply them together as one radical. If you multiply two identical square roots, remember that \(\sqrt{a} \cdot \sqrt{a} = a\). In our problem, we multiply:

• \(4 \sqrt{2}\) and \(3 \sqrt{12}\) gives \(12 \sqrt{24}\)
• \(4 \sqrt{2}\) and \(7 \sqrt{6}\) gives \(28 \sqrt{12}\)

To correctly multiply these, begin by multiplying the coefficients:

• \(4\) and \(3\) to get \(12\), and \(4\) and \(7\) to get \(28\)

Next, multiply the radicals:

• \(\sqrt{2} \cdot \sqrt{12} = \sqrt{24}\)
• \(\sqrt{2} \cdot \sqrt{6} = \sqrt{12}\)

Multiplying radicals is easier than it looks, as it breaks down into manageable steps. Just keep track of coefficients and radicals separately, and always simplify where possible.

Simplest Radical Form

Simplifying radicals involves writing them in their simplest form. The simplest radical form is achieved when no perfect square (other than 1) is a factor under the radical. Let's look at simplifying \(\sqrt{24}\) and \(\sqrt{12}\):

For \(\sqrt{24}\), we break it down as:

• \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\)

Next, for \(\sqrt{12}\), do the same:

• \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\)

Thus, after simplifying, \(12 \sqrt{24}\) becomes \(24 \sqrt{6}\), and \(28 \sqrt{12}\) becomes \(56 \sqrt{3}\). These are expressed in their simplest radical forms. Simplifying this way makes calculations easier and ensures you can compare or add similar radicals simply by checking their coefficients.