In Chapter 2, just after proving Proposition 2.3, we gave a cunning geometricalconstruction that demonstrated the existence of the real number √n for anypositive integer n. However, proving the existence of a cube root and, moregenerally, an nth root of any positive real number x is much harder and requiresa deeper analysis of the reals than we have undertaken thus far. We shall carryout such an analysis later, in Chapter 24. However, because we wish to includenth roots in the discussion of complex numbers in the next chapter, we pick outthe main result from Chapter 24 on such matters, namely Proposition 24.2, andstate it here. (It is, of course, proved in Chapter 24.) PROPOSITION 5.1 Let n be a positive integer. If x is a positive real number, then there is

exactly one positive real number y such that yn = x. If x,y are as in the statement, we adopt the familiar notation

y = x 1n . Thus, for example, 5 12 is the positive square root of 5, and 5 17 is the uniquepositive real number y such that y7 = 5.We can extend this notation to define rational powers of positive reals asfollows. Let x > 0. Integer powers xm (m ∈ Z) are defined in the familiar way:if m> 0 then xm = xx . . .x, the product of m copies of x, and x−m = 1xm ; and form = 0 we define x0 = 1.Now let mn ∈Q (with m,n ∈ Z and n≥ 1). Then we define

x mn = (x 1n)m . For example, 5− 47 is defined to be (5 17 )−4.