Problem 17 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(3 \sqrt{5}(2 \sqrt{2}-\sqrt{7})\)

Step-by-Step Solution

Answer

The product is \(6\sqrt{10} - 3\sqrt{35}\).

Step 1: Distribute the Scalar

First, distribute the number 3 to each term inside the parentheses. This means you'll multiply \(3\sqrt{5}\) by both \(2\sqrt{2}\) and \(-\sqrt{7}\).

Step 2: Multiply Radicals

When multiplying a scalar by a radical expression, you multiply the scalars together and the radicals together. So, \(3\sqrt{5} \times 2\sqrt{2}\) becomes \(6\sqrt{10}\) because \(3 \times 2 = 6\) and \(\sqrt{5} \times \sqrt{2} = \sqrt{10}\).

Step 3: Calculate the Second Component

Now multiply \(3\sqrt{5}\) by \(-\sqrt{7}\), resulting in \(-3\sqrt{35}\), because \(\sqrt{5} \times \sqrt{7} = \sqrt{35}\).

Step 4: Compose the Final Expression

Combine the results from Steps 2 and 3 to get the final expression: \(6\sqrt{10} - 3\sqrt{35}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, often represented as \( a(b + c) = ab + ac \). It allows us to multiply a single term by each term inside a set of parentheses. In our example, we start with the expression \(3 \sqrt{5}(2 \sqrt{2} - \sqrt{7})\). The distributive property tells us that we need to multiply \(3 \sqrt{5}\) by each of the terms \(2 \sqrt{2}\) and \(-\sqrt{7}\).
This step is crucial because it breaks down the multiplication into smaller, manageable parts. By distributing the scalar \(3 \sqrt{5}\), we create two separate multiplication problems that involve radicals, which can be handled with ease.

Multiplying Radicals

Multiplying radicals is another important algebraic operation. When we multiply radicals, we follow the principle that \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\). This allows us to combine the numbers under the square root when we multiply them. For example, in our main problem, the multiplication \(3\sqrt{5} \times 2\sqrt{2}\) becomes \(6\sqrt{10}\), because we calculate \(3 \times 2 = 6\) and \(\sqrt{5} \times \sqrt{2} = \sqrt{10}\).
We also need to pay attention to whether the radicals can be simplified or not, which will be explained further in the next section. In the second part of this multiplication, we handle \(3\sqrt{5} \times -\sqrt{7}\), which results in \(-3\sqrt{35}\), following the same multiplication principles.

Simplifying Square Roots

Simplifying square roots involves reducing the expression to its simplest form. While not all square roots can be simplified to integers, our goal is to break down the expression to make it easier to interpret or to identify if it can be reduced further.
For instance, after multiplying and obtaining \(6\sqrt{10}\) and \(-3\sqrt{35}\), check if the radicals \(\sqrt{10}\) and \(\sqrt{35}\) can be simplified. Simplification involves checking if the numbers under the square roots have any perfect square factors. In these examples, neither \(10\) nor \(35\) has factors that are perfect squares larger than 1, so they remain as they are.
When simplifying, always ensure no further extraction of square roots can be done. This step ensures that the final expression is as straightforward as possible.