= [(-1)^2 - (\sqrt7)^2] + [1 - (\sqrt7)^2] = (1 - 7) + (1 - 7) = -6 -6 = -12 - Decision Point
Simplifying the Expression: A Complete Guide to [(-1)² − (√7)²] + [1 − (√7)²] = −12
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](https://soloferat.biz.id/images/12---sqrt72--1---sqrt72--1---7--1---7---6--6---12-2439354.jpg)
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
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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:
