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Chapter 15: Problem 4

Explain why \(2 \sqrt{2}\) is in simplest form and \(\sqrt{8}\) is not in simplest form.

Short Answer

Expert verified
\(\sqrt{8}\) is not in simplest form because it can be simplified further into \(2 \sqrt{2}\). However, \(2 \sqrt{2}\) can't be simplified further, it's already in simplest form.

Step by step solution

01

Analyze \(2 \sqrt{2}\)

Inspect the expression \(2 \sqrt{2}\). As it stands, \(2 \sqrt{2}\) cannot be simplified further because \(\sqrt{2}\) is a surd, meaning it is an irrational number (cannot be expressed as a fraction of two integers) and can't be simplified by its square root.
02

Analyze \(\sqrt{8}\)

Next, inspect \(\sqrt{8}\). Now we know that 8 can be broken down into 2x2x2. We can simplify square roots by pulling out pairs of numbers. Doing this, you can pair two 2s together and pull them out of the square root, which results in \(2 \sqrt{2}\).
03

Final Thoughts

By simplifying \(\sqrt{8}\) we got the expression \(2 \sqrt{2}\). So, as you can see, \(\sqrt{8}\) wasn't in the simplest form, while \(2 \sqrt{2}\) already was.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Surds
A surd is a mathematical term that refers to an expression containing a root, such as a square root, which cannot be simplified to remove the square root symbol. In simpler terms, it is an expression involving an irrational number under a root that cannot be converted into an exact decimal or fraction. For example, the expression \( \sqrt{2} \) is a surd because its exact value cannot be expressed as a simple fraction. It remains in its root form.

This is why when you see an expression like \( 2\sqrt{2} \), the \( \sqrt{2} \) is considered a surd, and the expression cannot be simplified further because \( \sqrt{2} \) itself cannot be reduced any further.
Exploring Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a ratio of two integers, meaning they cannot be written as simple fractions. These numbers have non-repeating, non-terminating decimal expansions. Examples include numbers like \( \pi \), \( e \), and \( \sqrt{2} \).

In the context of simplifying square roots, irrational numbers often manifest within roots that cannot break down into integer ratios. For instance, \( \sqrt{8} \) contains \( \sqrt{2} \), an irrational number, which is why it originally wasn't in its simplest form. Through simplification, we focus on whether any simplification can eliminate the root expression involving an irrational number.
Using Pairing Factors to Simplify
When simplifying square roots, one effective method is using pairing factors. This involves breaking down the number under the square root into its prime factors and identifying pairs. Each pair of the same number can be 'pulled out' of the root.

Let's take \( \sqrt{8} \) as an example. You break down 8 into 2x2x2. By finding pairs, you pair two 2s and take one 2 out of the square root, simplifying into \( 2\sqrt{2} \). This method efficiently reduces the expression under the root by removing squared numbers, making the expression simpler.
Achieving the Simplest Form
The simplest form of a square root expression is achieved when all possible pairings of factors have been 'pulled out' of the root, and no further simplification is possible.

\( 2\sqrt{2} \) is already in its simplest form because the \( \sqrt{2} \) cannot be simplified further as there are no other factors to pair inside the root. This indicates that the expression cannot be reduced any further.

In contrast, starting with \( \sqrt{8} \), once you use the pairing method and arrive at \( 2\sqrt{2} \), you've reached the simplest form. Understanding this concept helps in ensuring that no further steps can reduce the expression, confirming it is fully simplified.

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