If $m + n = 24$, then find the value of

$$
(m - 16)^3 + (n - 8)^3.
$$

Question

If $m + n = 24$, then find the value of

$$
(m - 16)^3 + (n - 8)^3.
$$

Accepted Answer

We are given:

$$
m + n = 24
$$

Let’s simplify the expression:

$$
(m - 16)^3 + (n - 8)^3
$$

Let:

* $a = m - 16$
* $b = n - 8$

So the expression becomes:

$$
a^3 + b^3
$$

We know the identity:

$$
a^3 + b^3 = (a + b)^3 - 3ab(a + b)
$$

Now compute $a + b$:

$$
a + b = (m - 16) + (n - 8) = (m + n) - 24 = 24 - 24 = 0
$$

Now plug into the identity:

$$
a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 0^3 - 3ab \cdot 0 = 0
$$

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### **Final Answer:**

$$
\boxed{0}
$$

Answer

Answer

Answer