Question

How can you prove that the equation has a solution/is verifiable?

I've been given `-7+(16p+8)^(1/3)=-1` and I'm so confused

also in case it looks confusing, the 1/3 is an exponent thats a fraction, 16p+8 IS NOT over the 3

Answer 1

Yes, `p = 13`

Explanation:

Given:

`-7+(16p+8)^(1/3) = -1`

Adding `7` to both sides, this becomes:

`(16p+8)^(1/3) = 6`

Raising both sides to the power `3`, we find:

`16p+8 = 6^3 = 216`

Subtracting `8` from both ends, this becomes:

`16p = 208`

Dividing both sides by `16`, we find:

`p = 208/16 = 13`

With each of these steps, we performed the same operation on both sides of an equation. That means that if the equation before the steps holds then so will the equation after the step.

If the step is also reversible (e.g. adding `7` to both sides), then it follows that if the resulting equation holds so will the original.

If the step is not reversible, then it is possible that the resulting equation is true but the original is not.

The only step we might have doubts about is the one where we raise both sides to the power `3`. Note however that as a real valued function of real numbers, cubing is a one-to-one function. So this step is reversible too.

(Note that this would not be the case when raising both sides of an equation to an even power, since that is not reversible.)