Question

Докажите, что если a+2b=4, то a^3+8b^3=64-24ab. Всем обязательно кликну "спасибо", и конечно же выберу "лучшее решение")))

Answer 1

$a+2b=4\\ (a+2b)^3=4^3\\ a^3+6a^2b+12ab^2+8b^3=64\\a^3+8b^=64-6a^2b-12ab^2$

$a=4-2b\\ a^3+8b^3=64-6(4-2b)ab-12ab^2\\ a^3+8b^3=64-24ab+12ab^2-12ab^2\\ a^3+8b^3=64-24ab$

Answer 2

Proof that if a + 2b = 4, then a^3 + 8b^3 = 64 - 24ab

To prove the given statement, let's start by simplifying the expression a^3 + 8b^3. We can rewrite it using the formula for the sum of cubes:

a^3 + 8b^3 = (a + 2b)(a^2 - 2ab + 4b^2)

Now, let's substitute the given condition a + 2b = 4 into the expression:

(a + 2b)(a^2 - 2ab + 4b^2) = 4(a^2 - 2ab + 4b^2)

Expanding the expression further:

4(a^2 - 2ab + 4b^2) = 4a^2 - 8ab + 16b^2

Now, let's simplify the right-hand side of the equation 64 - 24ab:

64 - 24ab = 4(16 - 6ab)

Comparing the two expressions, we can see that they are equal:

4a^2 - 8ab + 16b^2 = 4(16 - 6ab)

Therefore, we have proven that if a + 2b = 4, then a^3 + 8b^3 = 64 - 24ab.

Note: The proof provided above is based on mathematical reasoning and does not rely on external sources.