Problem 4 - Intermediate Algebra

Question

Fill in the blanks. To __ \(\sqrt{24}\) means to write it as \(2 \sqrt{6}\)

Step-by-Step Solution

Answer

To simplify \(\sqrt{24}\), write it as \(2 \sqrt{6}\).

Step 1: Understanding the Problem

The problem requires us to simplify the square root of 24 and express it in the form similar to the given example, \(2 \sqrt{6}\). This involves breaking down \(\sqrt{24}\) into its simplest radical form.

Step 2: Factoring Inside the Square Root

First, factor the number 24. We identify that 24 can be written as the product of a perfect square number and another factor: \(24 = 4 \times 6\).

Step 3: Applying Square Root Property

Next, use the square root property which states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Applying this to \(\sqrt{24}\), we have \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6}\).

Step 4: Simplifying the Square Root

Now simplify \(\sqrt{4}\). Since 4 is a perfect square, \(\sqrt{4} = 2\). This gives us \(\sqrt{24} = 2 \times \sqrt{6}\).

Step 5: Completion

Finally, we can see that \(\sqrt{24}\) is indeed equal to \(2 \sqrt{6}\), matching the given fill-in-the-blank form.

Key Concepts

Simplest Radical Form

When simplifying square roots, we aim to express the square root in its simplest radical form. This means writing the square root of a number in the most reduced or simplified manner. To achieve this, we look for factors of the number under the root that include perfect squares. For example, simplifying \( \sqrt{24} \) involves breaking it down into \( \sqrt{4 \times 6} \). This step is crucial because 4 is a perfect square. Therefore, knowing how to identify and work with perfect squares helps us simplify square roots effectively.

Perfect Square Factorization

Perfect square factorization involves rewriting a number as the product of its perfect square and another factor. Perfect squares are numbers like 1, 4, 9, 16, and so on, because their square roots are integers. For instance, in the problem \( \sqrt{24} \), we recognize that 24 can be factored into \( 4 \times 6 \).
4 is a perfect square, but 6 is not. This kind of factorization is key in simplifying square roots because it helps us separate out the square root of a perfect square. Once separated, calculations become easier and more straightforward, allowing further simplification.

Square Root Property

The square root property is a mathematical rule that simplifies the handling of square roots. It states \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property is particularly useful when dealing with products of numbers inside square roots.
In our example, after factoring \( \sqrt{24} \) as \( \sqrt{4 \times 6} \), we can use the square root property. It allows us to break this into \( \sqrt{4} \times \sqrt{6} \). By applying the property, we simplify calculations as seen with \( \sqrt{4} = 2 \). This result simplifies \( \sqrt{24} \) into \( 2 \sqrt{6} \). Understanding this property is essential for anyone dealing with simplification of radicals.