Fix b>1. (a) If m, n, p, q are integers, n>0, q>0, and r=m /...

Fix $b>1$. (a) If $m, n, p, q$ are integers, $n>0, q>0$, and $r=m / n=p / q$, prove that $$\left(b^{m}\right)^{1 / n}=\left(b^{p}\right)^{1 / q} .$$ Hence it makes sense to define $b^{\prime}=\left(b^{m}\right)^{1 / n}$. (b) Prove that $b^{\prime+1}=b^{\prime} b^{\prime}$ if $r$ and $s$ are rational. (c) If $x$ is real, define $B(x)$ to be the set of all numbers $b^{t}$, where $t$ is rational and $t \leq x$. Prove that $$b^{\prime}=\sup B(r)$$ when $r$ is rational. Hence it makes sense to define $$b^{x}=\sup B(x)$$ for every real $x$. (d) Prove that $b^{x+y}=b^{x} b^{y}$ for all real $x$ and $y$.

Fix $b>1$.
(a) If $m, n, p, q$ are integers, $n>0, q>0$, and $r=m / n=p / q$, prove that $$\left(b^{m}\right)^{1 / n}=\left(b^{p}\right)^{1 / q} .$$
Hence it makes sense to define $b^{\prime}=\left(b^{m}\right)^{1 / n}$.
(b) Prove that $b^{\prime+1}=b^{\prime} b^{\prime}$ if $r$ and $s$ are rational.
(c) If $x$ is real, define $B(x)$ to be the set of all numbers $b^{t}$, where $t$ is rational and $t \leq x$. Prove that
$$b^{\prime}=\sup B(r)$$
when $r$ is rational. Hence it makes sense to define
$$b^{x}=\sup B(x)$$
for every real $x$.
(d) Prove that $b^{x+y}=b^{x} b^{y}$ for all real $x$ and $y$.

Key Concepts

Rational Exponentiation

This concept involves defining exponentiation when the exponent is a rational number. The definition is based on expressing the rational exponent as a quotient of integers. One proves that if two fractions represent the same rational number, then the corresponding expressions for exponentiation yield the same value. This ensures that the operation is well-defined, meaning that the value of b raised to any rational exponent does not depend on the particular representation chosen.

Properties of Exponents

This concept refers to the fundamental laws that govern exponentiation, such as the rule b^(x+y) = b^x * b^y. These properties are critical for enabling algebraic manipulations and for proving consistency when defining and extending exponentiation. The manipulation of exponents is used both to verify the definition of exponentiation for rational numbers and to support the extension of the operation to real exponents.

Supremum and Completeness of the Reals

This concept centers on the idea of a supremum, or least upper bound, which is guaranteed to exist for every bounded set of real numbers due to the completeness property of the real numbers. When extending exponentiation to real exponents, one considers a set of values corresponding to rational approximations and then defines the exponentiation result as the supremum of that set. This approach leverages the completeness axiom to ensure that the definition is valid for all real exponents.

Extension of Exponentiation to Real Exponents

This concept involves broadening the definition of exponentiation from rational exponents to all real exponents. By using rational approximations and the supremum concept, one defines b^x as the least upper bound of the set of values obtained from b raised to rational numbers that do not exceed x. This extension retains the fundamental properties of exponentiation, such as the multiplicative rule, and provides a rigorous foundation for dealing with real exponents.

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