Simplifying the Expression: A Complete Guide to [(-1)² − (√7)²] + [1 − (√7)²] = −12

Mathematics often involves unexpected twists — especially when exponents, square roots, and algebraic expressions combine in surprising ways. One such expression that commonly confuses beginners is:

[(-1)² − (√7)²] + [1 − (√7)²] = (1 − 7) + (1 − 7) = −6 − 6 = −12

Understanding the Context

At first glance, the equation appears complex, but with a step-by-step breakdown, it reveals elegant algebraic structure and straightforward simplification. In this article, we’ll explore the calculation, highlight key mathematical principles, and explain why this problem is a perfect example of applying order of operations, exponent rules, and square root properties in algebra.

Breaking Down the Expression Step-by-Step

Let’s begin with the full expression:
[(-1)² − (√7)²] + [1 − (√7)²]

= [(-1)^2 - (\sqrt{7})^2] + [1 - (\sqrt{7})^2] = (1 - 7) + (1 - 7) = -6 -6 = -12

Key Insights

We’ll simplify each bracketed term individually before combining them.

Step 1: Evaluate (-1)²

The square of a negative number follows the same rule as any real number:
 (-1)² = (−1) × (−1) = 1

So, the first bracket becomes:
 1 − (√7)²

Step 2: Simplify (√7)²

Continue Reading

\min |2\sin x + 3\cos x + 4| = \min (2\sin x + 3\cos x + 4) = 4 - \sqrt{13} + 0? \quad \text{Wait — no: the minimum of the inner expression is $ 4 - \sqrt{13} \approx 0.395 $, so the minimum of the absolute value is just its minimum value because it's positive.
\min |2\sin x + 3\cos x + 4| = 4 - \sqrt{13} \quad \text{if } 4 - \sqrt{13} > 0
But $ 4 - \sqrt{13} > 0 $ since $ \sqrt{13} < 4 $, so yes.

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Final Thoughts

By definition, squaring a square root cancels out:
 (√7)² = 7

Now the first bracket is:
 1 − 7 = −6

Step 3: Evaluate the Second Bracket [1 − (√7)²]

We already found (√7)² = 7, so:
 1 − 7 = −6

Step 4: Add Both Brackets Together

Now substitute both simplified brackets:
 (−6) + (−6) = −12

Thus:
[(-1)² − (√7)²] + [1 − (√7)²] = −6 + (−6) = −12

Key Algebraic Insights

This problem demonstrates several fundamental concepts: