%5E3%3D(2x)%5E3-7%5E3.jpeg "(2x-7)^3=(2x)^3-7^3")
The Flawed Logic: Why (2x-7)^3 ≠ (2x)^3 - 7^3
It's tempting to think that cubing a binomial like (2x - 7) is as simple as cubing each term separately. However, this is not how the distributive property works when dealing with exponents.
Let's break down why the equation (2x-7)^3 = (2x)^3 - 7^3 is incorrect.
Understanding the Difference
The equation (2x-7)^3 actually means (2x-7) multiplied by itself three times:
(2x-7)^3 = (2x-7)(2x-7)(2x-7)
To correctly expand this, we need to apply the distributive property. This involves multiplying each term in the first binomial by each term in the second binomial, then continuing the process with the third binomial.
The Correct Expansion
Here's how the correct expansion unfolds:
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Expand the first two binomials: (2x-7)(2x-7) = 4x^2 - 14x - 14x + 49 = 4x^2 - 28x + 49
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Multiply the result by the third binomial: (4x^2 - 28x + 49)(2x-7) = 8x^3 - 56x^2 + 98x - 28x^2 + 196x - 343
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Combine like terms: 8x^3 - 84x^2 + 294x - 343
Key Takeaway
The correct expansion of (2x-7)^3 is 8x^3 - 84x^2 + 294x - 343, which is not the same as (2x)^3 - 7^3.
This highlights the importance of carefully applying the distributive property and understanding how exponents work with binomial expressions.