A = \sqrt16(16 - 10)(16 - 10)(16 - 12) = \sqrt16(6)(6)(4) = \sqrt2304 = 48

Understanding the Algebraic Expression: A = √[16(16 – 10)(16 – 10)(16 – 12)] = 48

Calculus and algebra often intersect in powerful ways, especially when solving expressions involving square roots and polynomials. One such elegant example is the algebraic identity:

A = √[16(16 – 10)(16 – 10)(16 – 12)] = √[16 × 6 × 6 × 4] = √2304 = 48

Understanding the Context

This expression demonstrates a common technique in simplifying square roots, particularly useful in geometry, physics, and advanced algebra. Let’s break it down step-by-step and explore its significance.

The Expression Explained

We begin with:
A = √[16(16 – 10)(16 – 10)(16 – 12)]

$A = \sqrt{16(16 - 10)(16 - 10)(16 - 12)} = \sqrt{16(6)(6)(4)} = \sqrt{2304} = 48$

Key Insights

First, evaluate each term inside the parentheses:

(16 – 10) = 6
(16 – 12) = 4

So the expression becomes:
A = √[16 × 6 × 6 × 4]

Notice that (16 – 10) appears twice, making it a repeated factor:
A = √[16 × 6² × 4]

Now compute the product inside the radical:
16 × 6 × 6 × 4 = 16 × 36 × 4
= (16 × 4) × 36
= 64 × 36
= 2304

Hence,
A = √2304 = 48

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Final Thoughts

Why This Formula Matters

At first glance, handling nested square roots like √(a × b × b × c) can be challenging, but recognizing patterns simplifies the process. The expression leverages:

Factor repetition (6×6) to reduce complexity.
Natural grouping of numbers to make mental or hand calculations feasible.
Radical simplification, turning complex roots into clean integers.

Applications in Real-World Problems

This technique appears frequently in:

Geometry: Calculating diagonals or distances. For example, in coordinate geometry, √[a² + (a−b)² + (a−c)²] often leads to expressions similar to A.
Physics: Magnitude of vectors or combined forces, where perpendicular components multiply under square roots.
Algebraic identities: Helpful in factoring and solving quadratic expressions involving square roots.

How to Simplify Similar Expressions

If faced with a similar radical like √[x(a)(a − b)(x − c)], try:

Expand and simplify inside the root.
Look for duplicates or perfect squares.
Rewrite as a product of squares and square-free parts.
Pull perfect squares outside the radical.