Final answer:
To factor the sum of two cubes, x³+216, we identify it as a³ + b³ and apply the formula (a + b)(a² - ab + b²) to get (x + 6)(x² - 6x + 36). Cubing of exponentials involves multiplying the exponent by 3.
Step-by-step explanation:
Factoring the Sum of Two Cubes
The question involves factoring the sum of two cubes, specifically the expression x³+216. To factor a sum of two cubes, we use the formula a³ + b³ = (a + b)(a² - ab + b²). In our case, x³ can be seen as a³ and 216 can be translated to 6³ because 6 * 6 * 6 = 216. This results in the factorized form being (x + 6)(x² - 6x + 36).
Regarding the cubing of exponentials, when you cube a term like x², it becomes x¶ because you multiply the exponent by 3 (the cube). Similarly, when you deal with expressions like (3²)^5, you would end up with 3·, since the base remains the same and you multiply the exponents (2*5). This is based on the exponentiation rules, which dictate that (x^p)^q = x^(p*q).