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Chapter 8: Problem 37

Express each radical in simplified form. $$ \sqrt{12} $$

Short Answer

Expert verified
2 \( \sqrt{3} \)

Step by step solution

01

Identify factors of the radicand

The radicand is 12. Find the factors of 12 that include a perfect square. We have: 12 = 4 * 3, where 4 is a perfect square.
02

Apply the product rule for radicals

Using the product rule for radicals, \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we can rewrite \(\sqrt{12}\) as \(\sqrt{4 \cdot 3}\).
03

Simplify the radical expression

Simplify \(\sqrt{4 \cdot 3}\) to \(\sqrt{4} \cdot \sqrt{3}\). Since \(\sqrt{4} = 2\), the expression simplifies to 2 \(\sqrt{3}\).

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Key Concepts

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

radicals
A radical expression involves a root, such as a square root or cube root. The radicand is the number or expression inside the radical symbol, \( \sqrt{} \). For example, in \( \sqrt{12} \), the radicand is 12.
Radicals can often be simplified to make calculations easier and expressions clearer. The key is to factor the radicand into its prime components and identify any perfect squares. Understanding the components and structure of radical expressions is crucial for simplification and other operations.
product rule for radicals
The product rule for radicals is a useful tool for simplifying radical expressions. This rule states that \( \sqrt{a \, \cdot \, b} = \sqrt{a} \cdot \sqrt{b} \). This means you can split the square root of a product into the product of the square roots of the factors. Here鈥檚 how it works with our example, \( \sqrt{12} \):
  • First, factor the radicand (12) into its prime factors: 12 = 4 * 3.
  • Next, apply the product rule: \( \sqrt{12} = \sqrt{4 \cdot 3} \).
  • Then, separate it into two square roots: \( \sqrt{4} \cdot \sqrt{3} \).
  • Finally, simplify each part: \( \sqrt{4} = 2 \), so the expression becomes 2 \( \sqrt{3} \).
Using this rule helps break down complex expressions into simpler parts that are easier to manage.
perfect squares
Perfect squares are numbers that can be expressed as the square of an integer. Some common perfect squares are 1, 4, 9, 16, 25, and so on.
Recognizing perfect squares is helpful when simplifying radicals, as they can be perfectly resolved. For example, \( \sqrt{4} = 2 \) because 4 is a perfect square (2 * 2).
In our excercise, 12 was factored into 4 and 3 because 4 is a perfect square. By identifying and simplifying these perfect squares, you can make radical expressions easier to work with.

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