How to Simplify Expressions Involving Square Roots with Practical Steps

Dealing with square root expressions can be challenging, especially when there are complex terms involved. One problem in particular, √51 14√2 - √51 - 14√2, has been a point of confusion for many. This article will guide you through the process of simplifying such expressions through a step-by-step approach that leverages algebraic manipulation and properties of square roots.

Introduction to Square Roots

Square roots are a fundamental concept in algebra and are essential in various fields such as physics, engineering, and mathematics. Understanding how to simplify expressions involving square roots can greatly enhance one’s problem-solving abilities.

Simplifying Complex Square Root Expressions

1. Identifying and Rewriting Terms

Let's begin by defining the terms within the square roots. We are given the expression √51 14√2 - √51 - 14√2.

2. Squaring Both Sides to Simplify

Let a √51 14√2 and b √51 - 14√2. We aim to simplify a - b. Square both terms to remove the square roots: a^2 51 14√2 b^2 51 - 14√2 Calculate a^2 - b^2: a^2 - b^2 (51 14√2) - (51 - 14√2) 28√2 Use the difference of squares property: (a - b)(a b) a^2 - b^2 (a - b)(a b) 28√2 (a b) 2(√51 14√2 √51 - 14√2) 2(2√51) 4√51 This simplifies to:

a - b 28√2 / 4√51 2√2

3. Explanation of the Solution

The simplified form of the expression √51 14√2 - √51 - 14√2 is 2√2. This solution was derived using the principles of algebraic manipulation and the properties of square roots.

Verification and Validation

To verify the solution, we can square the simplified expression and check if it aligns with the original terms:

Let sqrt{51 14sqrt{2}} - sqrt{51 - 14sqrt{2}} y. Square both sides: y^2 (sqrt{51 14sqrt{2}} - sqrt{51 - 14sqrt{2}})^2 51 14sqrt{2} 51 - 14sqrt{2} - 2sqrt{(51 14sqrt{2})(51 - 14sqrt{2})} 102 - 2sqrt{2601 - 392} 102 - 2sqrt{2209} 102 - 94 8 This implies y ±2sqrt{2}

Alternative Solutions

One can also simplify the expression using alternative approaches:

1. Direct Simplification

Notice that:

sqrt{51 14sqrt{2}} - sqrt{51 - 14sqrt{2}} Let z sqrt{51 14sqrt{2}} - sqrt{51 - 14sqrt{2}} Then: z^2 (sqrt{51 14sqrt{2}} - sqrt{51 - 14sqrt{2}})^2 (51 14sqrt{2}) (51 - 14sqrt{2}) - 2sqrt{(51 14sqrt{2})(51 - 14sqrt{2})} 102 - 2sqrt{2601 - 392} 102 - 2sqrt{2209} 102 - 94 8 z ±2sqrt{2}

2. Further Simplification

This can be simplified to:

sqrt{51 14sqrt{2}} - sqrt{51 - 14sqrt{2}} 2sqrt{2}

Conclusion

Simplifying expressions involving square roots requires a combination of algebraic manipulation, the properties of square roots, and sometimes, a bit of creativity. By following a step-by-step approach, you can effectively simplify complex expressions like √51 14√2 - √51 - 14√2 to a more manageable form, such as 2√2.

Keywords

simplify square roots algebraic manipulation square root properties