Why Square Rooting Does Not Yield a b c in a^2 b^2 c^2

The equation a^2 b^2 c^2 is perhaps one of the most recognizable equations in mathematics, known as the Pythagorean theorem. It describes the fundamental relationship between the sides of a right-angled triangle, where a and b are the lengths of the two legs, and c is the length of the hypotenuse.

Square Rooting Both Sides

While the Pythagorean theorem is very powerful in its use, a common mistake often made is attempting to simplify the equation by taking the square root of both sides:

sqrt{a^2 b^2} sqrt{c^2}

This simplifies to:

sqrt{a^2 b^2} c

Why, despite this seeming simplification, does it not imply a b c?

Understanding the Operations

The operation of taking a square root is not a linear one. Importantly, the square root of a sum is not the sum of the square roots:

sqrt{a^2 b^2} neq sqrt{a^2} sqrt{b^2}

In fact:

sqrt{a^2 b^2} neq a b

This is true unless both a and b are zero. The inequality highlights the non-linear nature of square roots and why the initial assumption a b c is incorrect.

Geometric Interpretation

Geometrically, c is the length of the hypotenuse of a right triangle, while a and b are the lengths of the other two sides. The relationship a b c would imply that the two legs of the triangle are equal to the length of the hypotenuse, which is not true for any right triangle with non-zero sides. The relationship a^2 b^2 c^2 is a unique and pure relationship between the sides of a right triangle, not a direct sum of the lengths of the legs.

Real-World Implications

In the realm of real-world applications, the equation a^2 b^2 c^2 is not always true outside the context of right triangles. In situations where c is predetermined by the lengths of a and b, the equation is valid. However, when this predetermination is not present, the equation a^2 b^2 c^2 will only hold true by coincidence—most of the time, it does not.

Practical Examples

To see why this is the case, let's consider a simple example. If we take a 3 and b 4:

sqrt{3^2 4^2} sqrt{9 16} sqrt{25} 5

But, for the sake of comparison:

3 4 7

sqrt{3^2} sqrt{4^2} 3 4 7

Clearly, sqrt{3^2 4^2} neq 3 4 because the operation of taking the square root is not commutative with addition.

Conclusion

In summary, while the equation a^2 b^2 c^2 is valid for right triangles, square rooting both sides does not lead to a b c because the properties of square roots and the nature of the geometric relationship are inherently non-linear. The equation accurately represents the relationship between the sides of a right triangle, and any attempt to simplify it using linear operations will result in a misleading conclusion.

Understanding these principles is key to applying the Pythagorean theorem correctly and appreciating the limitations of mathematical operations when applied in a non-linear context.