Why Doesn't the Square Root of Sums Equal the Sum of Square Roots?

The common misconception that the square root of a sum equals the sum of the square roots of the individual terms is a frequent point of confusion in basic mathematics. This article aims to clarify why this is not true and provides an in-depth explanation with examples and mathematical proofs.

Introduction

Let's explore a specific example to understand the fundamental difference between the two expressions:

Take the equation:

[sqrt{a^2b^2} eq ab]

At first glance, it might seem intuitive, but to see why it's not true, let's walk through the steps:

Step-by-Step Mathematical Explanation

1. Start with the given equation:

[sqrt{a^2b^2} eq ab]

2. Square both sides of the equation:

[(sqrt{a^2b^2})^2 eq (ab)^2]

3. Simplify both sides:

[a^2b^2 eq a^2b^2]

4. Introduce a new term to the right-hand side:

[a^2b^2 eq a^2 cdot 2ab cdot b^2]

Here, the right-hand side shows that the square of the sum of square roots is more complex and includes an additional term (2ab) that is not present in the simplified form.

Exploration with Specific Values

To further illustrate, let's consider a specific scenario where (a b 1):

1. Substitute (a b 1) into the equation:

[sqrt{1^2 cdot 1^2} eq 1 cdot 1]

2. Simplify the left-hand side:

[sqrt{1} eq 1]

3. Take the square root of 1:

[sqrt{1} 1]

4. Observe that the right-hand side remains 1:

[1 eq 2sqrt{1}]

These steps clearly show that the square root of the sum is not equal to the sum of the square roots. In the example, (sqrt{1^2 cdot 1^2} 1), while (1 cdot 1 1) but does not equal (2sqrt{1} 2).

Generalization and Conclusion

The key to understanding the difference lies in the distribution of the square root operation over the product. The square root of a product is the product of the square roots, i.e., (sqrt{ab} sqrt{a} cdot sqrt{b}), but the square of a sum is not the sum of the squares:

[(a b)^2 eq a^2 b^2]

This is because the square of a sum involves an additional term, the cross term (2ab):

[(a b)^2 a^2 2ab b^2]

This term is what causes the discrepancy in the equation (sqrt{a^2b^2} eq ab).

In conclusion, the square root of the sum of two terms does not equal the sum of the square roots of those terms. This fundamental property is essential for understanding and applying mathematical concepts accurately in various contexts, from basic algebra to more advanced fields like calculus and statistics.