How many real nth roots does. . .. .? (I need help!)?

Question

How many real nth roots roots does a positive real number have if n is odd? If n is even?

That's a question on a worksheet. And you have as much info as I do.

Answer 1

If n is odd it has one.

If n is even it has two.

Take a look, for example at 1024.  It has two square roots(n=2), 32 and -32.  But only one cube root(n=3), which is approximately 10.  10*10*10 =1000, which is just under 1024.  But if you put in  -10, you're going to get -1000.  So -10 is not a cube root.

It is similar with higher roots, for example, 4^5 = 1024, whereas -4^5 = 1024.

Finally, (2^)10 = 1024 and (-2^)10=1024

Hope that helps

By the way, things get fun when you don't just consider real roots.  Every number has n nth roots.  That's, 2 square roots, 3 cube roots, 4 fourth roots and so on...

Most of these roots are complex numbers.  The cube roots of 8 are 2, 1+sqrt(3)*i, and 1-sqrt(3)*i.  Notice that the imaginary roots are the same except for the different sign of the imaginary part.  This gives a bit of a proof of the statement i gave about number of real roots, since if you have an odd number of roots an even number of them must be complex so there can only be 1 real one.