Simplify Square Root Of 112 In Radical Form

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Simplifying square roots can be a straightforward process that involves factoring the number under the square root sign into its prime factors and then taking the square root of as many pairs of identical factors as possible. In the case of simplifying the square root of 112 in radical form, we follow these steps:

Step 1: Factor 112 into Prime Factors

To simplify the square root of 112, we first break down 112 into its prime factors.

• Start by dividing 112 by the smallest prime number, which is 2.
• 112 ÷ 2 = 56
• 56 ÷ 2 = 28
• 28 ÷ 2 = 14
• 14 ÷ 2 = 7

So, the prime factorization of 112 is 2^4 * 7.

Why Prime Factorization is Important

Prime factorization is crucial in simplifying square roots because it allows us to identify perfect squares that can be taken out of the square root.

Step 2: Identify Perfect Squares and Simplify

Given the prime factorization of 112 as 2^4 * 7, we look for pairs of identical factors. Here, we have 2^4, which can be broken down into 2^2 * 2^2, or (2^2)^2.

• Therefore, the square root of 112 can be simplified as the square root of (2^2)^2 * 7.
• This simplifies to 2^2 * square root of 7.

Simplified Radical Form

Expressing this in radical form, we have:

√112 = 4√7

This means that the simplified radical form of the square root of 112 is 4 times the square root of 7.

Key Points

• Prime Factorization: Breaking down a number into its prime factors is essential for simplifying square roots.
• Identifying Perfect Squares: Recognizing pairs of identical prime factors allows us to take the square root of those factors.
• Radical Form: The simplified form of the square root of 112 in radical form is 4√7.

Conclusion and Further Applications

Simplifying square roots into radical form is a fundamental skill in mathematics, used in various algebraic manipulations, geometric calculations, and problem-solving. The process involves breaking down numbers into their prime factors, identifying perfect squares, and simplifying accordingly.

For those interested in further applications, simplifying square roots is crucial in solving quadratic equations, understanding the properties of parabolas in geometry, and in many real-world applications involving measurement, physics, and engineering.