Step 1: Identify the Algebraic Pattern
 

Let a = 1.55
Let b = 0.55

By substituting these variables, the expression becomes:

a² + b² + ab

Step 2: Recall the Relevant Algebraic Identity
The pattern a² + ab + b² is a key component of the "difference of cubes" identity. The full identity is:

a³ − b³ = (a − b)(a² + ab + b²)

Step 3: Apply the Identity to the Expression
From the identity, we can see that our expression `(a² + ab + b²)` can be isolated by dividing both sides by `(a - b)`:

(a² + ab + b²) = (a³ − b³) / (a − b)

To proceed, we must calculate the value of `(a - b)`.

Step 4: Calculate the Value of (a − b)
Using the values we assigned in Step 1:

a − b = 1.55 − 0.55 = 1

Step 5: Arrive at the Final Solution
Now, substitute the result from Step 4 back into the equation from Step 3:

(a² + ab + b²) = (a³ − b³) / 1

This simplifies to:

a² + ab + b² = a³ − b³

Conclusion

By substituting the original numbers back into the simplified equation, we have proven that the initial expression is indeed equal to the difference of their cubes:

1.55² + 0.55² + (1.55 × 0.55) = 1.55³ − 0.55³