Describe why simplifying square roots into their most reduced radical form is important in mathematical problems solving also outlined the steps involved in simplifying a square root to its simplest radical form

Answer 1

Simplifying square roots into their most reduced radical form is important in mathematical problem-solving for several reasons:

1. Clarity: Simplifying square roots makes expressions clearer and easier to understand. This clarity is particularly helpful when communicating solutions or working through multi-step problems.

2. Easier Calculations: Simplified radicals can reduce the complexity of calculations. In subsequent mathematical operations—such as addition, subtraction, or multiplication—working with simplified forms can lead to more straightforward computations.

3. Error Prevention: When radicals are in their simplest form, it reduces the risk of errors in calculations. Simplifying helps ensure that all factors are accounted for, preventing mistakes that can arise from handling complex, unsimplified roots.

4. Forms Consistency: Many mathematical conventions prefer or require that answers be presented in their simplest radical form, ensuring consistency across solutions and making them easier to compare.

5. Facilitates Further Mathematical Operations: In more advanced mathematics, simplified forms allow for easier integration, differentiation, and substitutions, especially in calculus and algebra.

Steps to Simplify a Square Root to Its Simplest Radical Form:

To simplify a square root, follow these steps:

1. Factor the Number Under the Radical: Identify the number under the square root and factor it into its prime factors. For example, for \(\sqrt{72}\), factor it into \(72 = 36 \times 2\) or \(72 = 2^3 \times 3^2\).

2. Identify Perfect Squares: Look for any perfect squares in the factors. Perfect squares are numbers whose square root is an integer (e.g., \(1, 4, 9, 16, 25, 36, etc.\)). In our example, \(36\) is a perfect square.

3. Take Out the Square Root of the Perfect Square: For each perfect square factor, take the square root and move it outside the radical. Continuing with \(\sqrt{72}\):

   \[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \]

4. Check for More Perfect Squares: Ensure that there are no more perfect square factors left under the radical. If any exist, repeat the first three steps until you cannot simplify further.

5. Final Result: Write the final simplified form as the product of any factors pulled out of the square root and the remaining radical. The final result for our example is \(6\sqrt{2}\).

In summary, simplifying square roots enhances the clarity and efficiency of mathematical problem-solving and is an essential skill for students and professionals alike in mathematics and related fields.