Question

Explain why it is sometimes necessary to express square roots in their simplest radical form. Then, explain the process of rewriting a square root in its simplest radical form.(2 points)

Answer 1

Expressing square roots in their simplest radical form is often necessary for several reasons:

1. **Clarity and Precision**: Simplest radical form provides a clearer understanding of the exact value represented by the square root. It is easier to work with and interpret than a decimal approximation, which can be less precise and harder to manage in calculations.

2. **Mathematical Conventions**: Certain mathematical procedures and rules are easier to apply when expressions are in their simplest form. For example, simplifying square roots can aid in addition, subtraction, and rationalizing denominators, making it easier to combine terms or solve equations.

The process of rewriting a square root in its simplest radical form involves the following steps:

1. **Factor the Number Inside the Square Root**: Identify the number under the square root and factor it into its prime factors or into a product of squares. For example, if you have √72, you can factor it as √(36 × 2).

2. **Simplify Using Perfect Squares**: Look for perfect squares (e.g., 1, 4, 9, 16, 25, etc.) among the factors. The square root of a perfect square can be simplified. Continuing the example, √(36 × 2) can be simplified to √36 × √2, which equals 6√2.

3. **Combine Factors**: Express the final result as a product of whole numbers and any remaining square roots that cannot be simplified further. The expression should be in the form of \(a\sqrt{b}\), where \(a\) is a whole number and \(b\) is not divisible by a perfect square other than 1.

By following these steps, you can effectively rewrite square roots in their simplest radical form, making mathematical expressions clearer and easier to work with.