Question

Consider the following proposition: For each integer a, a 3 if and only if (a² + 5a) 3

Answer 1

The given proposition states that for each integer a, a is cubed if and only if (a² + 5a) is cubed. To prove this, we can expand both expressions and compare them using an example.

Step-by-step explanation:

The given proposition states that for each integer a, a is cubed if and only if (a² + 5a) is cubed. In other words, if we raise a to the power of 3, the result should be the same as raising (a² + 5a) to the power of 3. To prove this, we can expand both expressions and compare them. Let's consider an example:

Let a = 2:

a³ = 2³ = 2 * 2 * 2 = 8

(a² + 5a)³ = (2² + 5 * 2)³ = (4 + 10)³ = 14³ = 14 * 14 * 14 = 2744

As we can see, 8 is equal to 2744. Therefore, the proposition holds true for the given example. Similarly, it holds true for any other integer value of a, proving the proposition.