Fix b > 1. If m, n,(DUE IN 1 HOUR)

[Jamin2112]

Fix b > 1. If m, n, ... (DUE IN 1 HOUR!)

Homework Statement

Fix b > 1.

If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that
(b^m)^1/n = (b^p)^1/q

Homework Equations

For every real x > 0 and every integer n > 0 there is one and only one positive real y such that yⁿ = x.

The Attempt at a Solution

This is what I have so far.

I could use a helpful hint or two to put the final nail in this coffin.

 

[mighty2000]

it is important to approach problems systematically and logically. In this case, we are asked to prove that (bm)1/n = (bp)1/q. We can start by considering the definition of a positive real number raised to a power. According to the given equation, we know that (bm)1/n is equal to x, where x is a positive real number. Similarly, (bp)1/q is also equal to x.

Next, we can use the given information that r = m/n = p/q. This means that m = rn and p = rq. Substituting these values into our original equation, we get (brn)1/n = (brq)1/q.

Now, we can use the property of exponents that (ab)n = anbn to rewrite this equation as (br)n/n = (br)n/q. By the definition of a positive real number raised to a power, we know that (br)n/n is equal to br and (br)n/q is equal to bp. Therefore, we can conclude that br = bp, which proves that (bm)1/n = (bp)1/q.

To summarize, we started by considering the definition of a positive real number raised to a power and then used the given information to manipulate the equation and arrive at our desired result. This proof can be considered complete, but as a scientist, it is always important to double check our work and make sure all steps are logical and valid.