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the Roots of Rational Numbers |
The Unique Factorization Theorem for Integers, which is often called, The Fundamental Theorem of Arithmetic, says that any integer can be expressed as a unique product of powers of primes. Unique means that there is only one set of primes and their exponents whose product gives the integer. This theorem is proven, among other places, at Unique Factorization.
Suppose Q is a rational number expressed in its reduced form as g/h where g and h have no common factor. Then unless there exists a rational number r/s such that Q=r^{k}/s^{k} then Q^{1/k} is irrational. In other words, unless Q is the ratio of two k-th powers of integers Q^{1/k} is irrational. This takes care of proving that the irrationality of numbers such as the cube root of 3 or the fourth root of (1/2) and so forth.
Proof: Suppose Q^{1/k} is the rational number n/m, where n and m have no common factor. The condition for Q is one of if and only if. The if part is obvious. The only if part just requires proving that the ratio of k-th powers is a reduced form.
Suppose Q is the rational number. If Q^{1/k} then
Since n and m have no common factor, n^{k} and m^{k} have no common factor. This means n^{k}/m^{k} is a reduced form for Q. There is only one reduced for for Q, hence n^{k}/m^{k} is it.
Thus if a rational number Q is not of the form n^{ k}/m^{k} then Q^{1/k} is irrational.
Consider a term R of the form P^{1/h}Q^{1/k} where P and Q are rational numbers and h and k are positive integers. Then
If R^{hk} is not the ratio of hk powers of integers then it hk-th root, namely R, is irrational.
For example, suppose R=(2)^{1/3}(5)^{1/2}. Then R^{6} = 2^{2}5^{3}, which is equal to 4·125=500 or 500/1. Since 500 is not the sixth power of any integer, R is irrational.
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