Multiple solutions or one or none?

Question

Solve the equation showing all your work:

Solve the equation showing all your work:

Which of the above equations has multiple solutions? Explain why.

Answer 1

So let's start with the first equation you're given, (3x-7)^5/2 = 32.

Any number/expression raised to a fraction becomes a radical expression, with the denominator being the index (nth root) and the numerator being the exponent that the radicand (expression inside the radical) is raised to. In this case, that would look like this:

√(3x-7)⁵ = 32.

However, when you simplify that expression, you'd get (3x-5)²√(3x-5) = 32, which is pretty unpleasant to deal with. So what we'll do instead is this:

(3x-5)^(5/2)(2/5) = 32^(2/5)

This undoes the exponent on the left side of the equation, and makes the right side much more manageable to deal with. So we're left with this:

3x - 7 = 32^(2/5), which we can simplify into the 5th root of 32² (this text box won't let me do anything higher than a square root). When you calculate that, you get 4.

But anyways, we can still solve that algebraically, so we would ultimately get the expression x = (4 + 7) ÷ 3. You'll get 11/3, which is 3.6 repeating. You only get one solution.

So now onto the second equation: 16x^(2/5) = 25, we can solve it in either way. I'm going to do it this way:

16x^(2/5)(5/2) = 25^(5/2)

We're left with 16x = 25^(5/2), which translates into √(25)⁵. This is nice because 25 already has a square root (5) so when we simplify that radical expression, we can turn it into √(5²)⁵, or √5¹⁰. Upon simplifying that, you get ±3,125, which you then divide by 16, and get ±195.3125. Since that one can be positive or negative, it brings you two solutions.