Intereting Posts

What number appears most often in an $n \times n$ multiplication table?
Showing $\{a+b\sqrt{2} \in R$ | $a$ is divisible by $2\}$ is an ideal.
An irreducible $f\in \mathbb{Z}$, whose image in every $(\mathbb{Z}/p\mathbb{Z})$ has a root?
Separable First Order Ordinary Differential Equation with Natural Logarithms
Estimating a probability of head of a biased coin
Designing an Irrational Numbers Wall Clock
T$\mathbb{S}^{n} \times \mathbb{R}$ is diffeomorphic to $\mathbb{S}^{n}\times \mathbb{R}^{n+1}$
second derivative at point where there is no first derivative
geometrical interpretation of a line integral issue
How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p$?
A surjective homomorphism between finite free modules of the same rank
Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way
How do I calculate $I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$?
Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$
Courses on Homotopy Theory

For all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$, Prove that $a^b\cdot a^c=a^{b+c}$

I have come across this question and its bugging me. Its a basic property that we learn in HS and I was hoping someone can enlighten me

- Confused about complex numbers
- Growth of exponential functions vs. Polynomial
- Solving Induction $\prod\limits_{i=1}^{n-1}\left(1+\frac{1}{i}\right)^{i} = \frac{n^{n}}{n!}$
- Non-integral powers of a matrix
- Why is the math for negative exponents so?
- What is the solution to the equation $9^x - 6^x - 2\cdot 4^x = 0 $?

- Solve the equation. e and natural logs
- Non-integer powers of negative numbers
- How to evaluate fractional tetrations?
- Modular Exponentiation 3^5^7
- Prove that $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$
- A question comparing $\pi^e$ to $e^\pi$
- How do I understand $e^i$ which is so common?
- Why is $0^0$ also known as indeterminate?
- Fastest way to check if $x^y > y^x$?
- Whether matrix exponential from skew-symmetric 3x3 matrices to SO(3) is local homeomorphism?

On way of doing this is to define $a^x$ for real $x$ using the least upper bound property of the real numbers.

$$ a^x = \text{ l.u.b of } \lbrace a^t \mid t< x, t\in \mathbb{Q} \rbrace $$

This means we consider the values for all rational powers which are less than $x$ and define the result of raising to the $x$ power as the smallest real number that is greater than or equal to the elements of this set.

To prove the product rule then we can first look at the meaning of $a^xa^y$ when $x,y\in \mathbb{R}$. Consider the following set,

$$ B= \lbrace a^r a^t \mid r < x, t < y, r \in \mathbb{Q}, t \in \mathbb{Q} \rbrace $$

We can conclude the following,

$$ a^r < a^x \text{ by the definition of } a^x.$$

$$ a^t < a^y \text{ by the definition of } a^y.$$

$$ \text{ Therefore } a^r a^t < a^x a^y$$

I am going to leave the proof that $a^xa^y$ is the l.u.b of $B$ to the thoughtful reader.

We can now look at $a^{x+y}$, this is by definition

$$ a^{x+y} = \text{l.u.b of } \lbrace a^s \mid s < x+ y, s \in \mathbb{Q} \rbrace = \text{ l.u.b of } \lbrace a^{r+t} \mid r < x, t < y, r \in \mathbb{Q}, t \in \mathbb{Q} \rbrace. $$

Using our knowledge of the power rule for rational number (which is a different proof) we notice that this new set is really just $B$. Therefore $a^{x+y}$ is just the l.u.b of $B$. A set cannot have two distinct least upper bounds therefore $a^{x+y}=a^xa^y$.

Notice that there is no need to appeal to the exponential function (which is far easier) to establish the existence and properties of the real powers. All that is necessary is the defining property of the real numbers which distinguishes them from the rationals (the least upper bound property).

For rational powers we have to take a different approach. We first define rational powers using $n$th roots. If $m,n \in \mathbb{Z}$ then define,

$$ a^{m/n} \equiv \sqrt[n]{a^m} .$$

Now suppose we multiply two expressions with different rational powers,

$$ a^{m/n} a^{p/q} = \sqrt[n]{a^m} \sqrt[q]{a^p} .$$

We will label the left hand side with the variable $K$ giving us,

$$ K = \sqrt[n]{a^m} \sqrt[q]{a^p} .$$

Taking the $n\cdot q$ power of both sides (remember $n$ and $q$ are integers so this is well defined) we get,

$$ K^{nq} = \left(\sqrt[n]{a^m} \sqrt[q]{a^p}\right)^{nq}$$

$$ K^{nq} = \left(\sqrt[n]{a^m} \right)^{nq} \left( \sqrt[q]{a^p}\right)^{nq}$$

$$ K^{nq} = \left[ \left( \sqrt[n]{a^m} \right)^n\right]^{q} \left[ \left(\sqrt[q]{a^p} \right)^q \right]^{n}$$

$$ K^{nq} = \left(a^m \right)^{q} \left( a^p\right)^{n}$$

$$ K^{nq} = a^{mq} a^{pn} $$

$$ K^{nq} = a^{mq+pn} $$

Take note that in every step above I only either used the rules of exponents for integers or the definition of $n$’th roots.

Now we will remove the power from $K$ using radicals.

$$ K = \sqrt[nq]{a^{mq+pn}} $$

Recalling the definition of rational powers we rewrite the right hand side as,

$$ K = a^{\frac{mq+pn}{nq}} = a^{\frac{m}{n} + \frac{p}{q}}$$

Replacing $K$ with its value we have,

$$a^{m/n} a^{p/q} = a^{\frac{m}{n} + \frac{p}{q}}$$

and therefore we have established the product rule for exponents.

We give a formal argument. It will not be entirely easy.

Let’s assume that we have defined the exponential function $\exp(t)=e^t$ in one of the many ways it can be done, and proved that its derivative is $e^t$. We define $a^x$ as $\exp(x\ln a)$.

For simplicity, write $k$ instead of $\ln a$. We want to prove that $\exp(ku)\exp(kv)=\exp(k(u+v))

$.

Keep $u$ fixed, and let

$$f(v)=\frac{\exp(k(u+v))}{\exp(kv)}.\tag{1}$$

Differentiate with respect to $v$. We get by the Quotient Rule and the Chain Rule

that $f'(v)=0$. (If you need details here, they can be supplied.)

So $f(v)$ is a constant function. Which constant?

Set $v=0$. We find that our constant is $\exp(ku)$. Now (1) gives us the result.

- Show that if $x\geq 0$ and $n$ is a positive integer, then $\sum_{k=0}^{n-1}\left\lfloor {x+\frac{k}{n}}\right\rfloor=\lfloor {nx}\rfloor$
- Is there a possibility to choose fairly from three items when every choice can only have 2 options
- A three-way duel (probability puzzle)
- Is there a branch of mathematics that requires the existence of sets that contain themselves?
- Orientation of Boundary of Lower Hemisphere – Stokes's Theorem
- Intuition for Smooth Manifolds
- Prove that the set of positive rational values that are less than $ \sqrt{2}$ has no maximum value
- Two equivalent definitions of a.s. convergence of random variables.
- Example of two open balls such that the one with the smaller radius contains the one with the larger radius.
- There is a bijection between irreducible components of the generic fiber and irreducible components passing through it.
- Other Algebraically Independent Transcendentals
- What REALLY is the modern definition of Euclidean Spaces?
- Prove that bitstrings with 1/0-ratio different from 50/50 are compressable
- Find the last two digits of $9^{{9}^{9}}$
- What is the difference between an indefinite integral and an antiderivative?