# Proof real exponentiation

Let x,y > 0 be positive reals, and let n, m >= 1 be positive integers,

Prove:

If x > 1, then x ^ 1/k is a decreasing function of k. If x < 1, then x ^ 1/k is an increasing function of k. If x=1, then x ^ 1/k = 1 for all k.

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#### Solution Preview

Let x,y > 0 be positive reals, and let n, m >= 1 be positive integers,

Prove:

If x > 1, then x ^ 1/k is a decreasing function of k. If x < 1, then x ^ 1/k is an increasing function of k. If x=1, then x ^ 1/k = 1 for all k.

We recall the definition of an increasing function.

A function f is said to be an ...

#### Solution Summary

The expert examines proof real exponentiation.

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