a^3 + b^3 = 7^3 - 3 \times 10 \times 7 = 343 - 210 = 133 - Redraw
Understanding the Equation: a³ + b³ = 7³ − 3 × 10 × 7 = 343 − 210 = 133
Understanding the Equation: a³ + b³ = 7³ − 3 × 10 × 7 = 343 − 210 = 133
Mathematical expressions often hide elegant relationships and surprising simplifications, and the equation a³ + b³ = 7³ − 3 × 10 × 7 = 343 − 210 = 133 is a perfect example. While it begins deceptively straightforward, this equation reveals a fascinating interplay of cubes, arithmetic operations, and numerical simplification.
In this breakdown, we’ll explore how a³ + b³ equals 7³ − 3 × 10 × 7, ultimately computing to 133, a number rich in mathematical interest.
Understanding the Context
Breaking Down the Left-Hand Side: a³ + b³
On the left side, the identity a³ + b³ is a classic algebraic form known as the sum of cubes. This can be factored as:
$$
a³ + b³ = (a + b)(a² - ab + b²)
$$
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Key Insights
While we won’t use the factorization here, recognizing this identity helps frame the relationship between a and b, especially when linked to specific numerical values or geometric interpretations.
Examining the Right-Hand Side: 7³ − 3 × 10 × 7
The right-hand side starts with 7³, meaning 343:
$$
7³ = 343
$$
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Then subtracts 3 × 10 × 7, which calculates to:
$$
3 × 10 × 7 = 210
$$
So the expression becomes:
$$
343 - 210 = 133
$$
This step demonstrates a direct arithmetic reduction—simple subtraction based on precise multiplication and exponentiation.
Connecting Both Sides: a³ + b³ = 133
Now, we know:
$$
a³ + b³ = 133
$$
The challenge becomes finding integers a and b such that their cubes sum to 133. While multiple real number pairs satisfy this, often the problem implies integer solutions for instructional clarity.
