## Understanding the (a² + b² + c²)² Formula

The formula (a² + b² + c²)² is a simple yet powerful algebraic expression that finds applications in various mathematical fields, particularly in geometry and trigonometry. It describes the square of the sum of squares of three variables, a, b, and c.

### Expanding the Formula

Expanding the formula (a² + b² + c²)² involves using the distributive property of multiplication. We can rewrite it as:

(a² + b² + c²)² = (a² + b² + c²)(a² + b² + c²)

Now, multiplying each term in the first bracket with each term in the second bracket, we get:

(a² + b² + c²)² = a⁴ + a²b² + a²c² + b²a² + b⁴ + b²c² + c²a² + c²b² + c⁴

This simplifies to:

**(a² + b² + c²)² = a⁴ + b⁴ + c⁴ + 2a²b² + 2a²c² + 2b²c²**

### Geometric Interpretation

The formula (a² + b² + c²)² can be visualized geometrically. Imagine a rectangular box with sides of length a, b, and c. The square of the sum of squares, (a² + b² + c²)², represents the square of the diagonal of this box. This diagonal can be calculated using the Pythagorean theorem, as the square root of the sum of squares of the sides.

### Applications

The (a² + b² + c²)² formula has various applications, including:

**Finding the diagonal of a rectangular box:**As mentioned earlier, the formula directly relates to the diagonal of a rectangular box.**Calculating distances in three-dimensional space:**This formula is essential for calculating distances between points in three-dimensional space.**Solving problems in trigonometry:**The formula can be used to derive trigonometric identities and solve trigonometric equations.

### Conclusion

The (a² + b² + c²)² formula provides a concise and elegant expression for the square of the sum of squares of three variables. Its simplicity and versatility make it a valuable tool in various mathematical disciplines, particularly geometry and trigonometry. Understanding this formula allows us to solve complex problems involving distances, volumes, and other geometric concepts.