Final answer:
To factor completely using the formula for the sum of cubes, rewrite the expression as (a^3 - b^3) and use the formula (a^3 - b^3) = (a - b)(a^2 + ab + b^2). Simplify the expressions to obtain the completely factored form.
Step-by-step explanation:
To factor completely using the formula for the sum of cubes, we can write the expression as (a^3 - b^3). In this case, 643 is the cube of 17 (17^3) and 125 is the cube of 5 (5^3). So we can rewrite the expression as (17^3 - 5^3).
Using the formula for the sum of cubes, (a^3 - b^3) = (a - b)(a^2 + ab + b^2), we can factor out the expression as (17 - 5)(17^2 + 17*5 + 5^2).
Simplifying further, (17 - 5) = 12 and (17^2 + 17*5 + 5^2) = 289 + 85 + 25 = 399. Therefore, the completely factored expression is 12 * 399 = 4788.
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