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 Simplicity: Simplifying square roots can make mathematical expressions clearer and easier to read. It often helps in identifying relationships between numbers, especially in algebra and geometry.

2. Easier Calculations: When square roots are in their simplest form, they are more manageable for further calculations. This can be particularly important in solving equations, simplifying expressions, or performing operations like addition or subtraction.

Process of Rewriting a Square Root in Its Simplest Radical Form:

1. Identify the Square Root: Start with a square root expression, such as \( \sqrt{n} \), where \( n \) is a positive integer.

2. Factor the Number: Factor \( n \) into its prime factors or as a product of a perfect square and another number. For example, if \( n = 50 \), it could be factored as \( 25 \times 2 \) (since \( 25 \) is a perfect square).

3. Apply the Square Root Property: Use the property of square roots that states \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). In our example, \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} \).

4. Simplify: Evaluate the square root of any perfect squares. Continuing from our example, \( \sqrt{25} = 5 \).

5. Combine and Write in Simplest Form: Combine the results to express the square root in simplest form. In our case, \( \sqrt{50} = 5\sqrt{2} \).

By following these steps, you can express square roots in their simplest radical form, making them easier to work with in further mathematical contexts.