Please solve these radical equations and show the steps so that I can understand them.

These are expressions not equations, so with the exception of (4) can only be simplified at best. So those instructions are therefore not applicable here.

I don't know what you mean by "extraneous" solutions but equations with x^(2 and over) can have what are called their "roots", which I think are where y = 0. When plotted as graphs these are the curves' intercepts on the x-axis.

However, if it's any help, and I am stretching what algebra I can recall and understand, I offer these. Sorry - I can't see how to type indices correctly in what's only a basic text-editor.

(1) simplify first to 4√n * 3√n.

That is because the original end n term is the square root of n squared.

I think it then becomes

[12√n√n] = 12n.


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2) A reciprocal index signifies the root. So the expression is the

[(cube-root of x) / (6th root of x)],

but I'm afraid I'm stuck at that point. It might simplify further but I can't see how.

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3) I think that is only the square root of [(5√x)] so doesn't really get you anywhere. You may as well just replace the index at the end with a square-root sign at the front.

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4) This one does simplify to a number because the index is working on a definite value; and one with a nice simple root. The index (1/2) means the square root. Making the index negative means the root's reciprocal. So:

[36^(- 1/2)] = [1/(√36)] = 1/6 = 0.1666 recurring.

(I had to verify this by calculator.... )