Solving for the Square Root of 105^3 - 104^3: A Detailed Approach

In this article, we explore the detailed steps needed to calculate the square root of 105^3 - 104^3, an intriguing problem that involves both algebraic identities and arithmetic operations. This exercise will help understand the application of algebraic identities, the cube difference formula, and practical estimation techniques.

Introduction

The problem at hand is to compute the square root of 105^3 - 104^3. This expression can be broken down and solved using algebraic manipulation and fundamental mathematical identities. The solution will be derived step-by-step, providing a clear and detailed explanation for each step.

Step-by-Step Solution

Step 1: Use the Difference of Cubes Identity

The difference of cubes formula is given by:

A^3 - B^3 (A - B)(A^2 AB B^2)

In our case, A 105 and B 104. Thus, we can rewrite the expression as:

[105^3 - 104^3] [104 1^3 - 104^3] [104^3 3104^21 31041^2 1^3 - 104^3] [3104^2 3104 1]

Step 2: Simplify the Expression

By simplifying the expression further, we get:

[104^3 3104^2 3104 1 - 104^3] [3104^2 3104 1] [3104{104 1} 1] [3104105 1] [310920 1] [32760 1] 32761

This simplifies to 32761, which is the value under the square root.

Step 3: Calculate the Square Root

Now, we need to find the square root of 32761:

sqrt{105^3 - 104^3} sqrt{1157625 - 1124864} sqrt{32761} 181

Therefore, the square root of 105^3 - 104^3 is 181.

Alternative Method: Factoring and Simplification

An alternative approach involves factoring the expression:

A^3 - B^3 (A - B)(A^2 AB B^2)

Here, A 105 and B 104, so:

[105^3 - 104^3] [105 - 104][105^2 104 105 104^2] [1][105^2 10920 104^2] [105^2 10920 104^2] [11025 10920 10816] 32761

Thus, sqrt{105^3 - 104^3} sqrt{32761} 181

Estimation and Verification

We can also estimate the square root by considering the physical interpretation of the problem. Imagine a cube of 104 units on each side. To extend this to a 105-unit cube, we add:

104^2 10816 square units on three non-opposite faces. 3 x 104 312 units along the edges where these faces meet. 1 unit at the corner where the three faces meet.

The sum of these units is:

32761

The square root of 32761 is 181, confirming our previous calculations.

Conclusion

This detailed exploration of the problem not only highlights the application of algebraic identities and simplifications but also provides a practical approach to solving such problems. The step-by-step method and alternative techniques offer valuable insights into how to tackle similar mathematical challenges.