The concept of exponent, as said, began with Archimedes, who when studying volumes, obtained a very, very large number as said in another introductory power post. In this post, we will talk about powers for negative exponents.

Negative Exponents

On this site we talk about power with an integer exponent, however, we do not talk about negative exponents, as the result is a rational number, which was part of the context.

Grafiko's Galileo.

The base of the power can be any number. In this post we will use rational numbers:

$$\mathbb{Q} = \left\\{\dots, -3, -\dfrac{5}{2},-2, -\dfrac{3}{2} ,-1, -\dfrac{1}{2},0 , \dfrac{1}{2} , 1, \dfrac{3}{2} , 2, \dfrac{5}{2} , 3, \dots\right\\}$$

The exponent, in this context, is an integer.

$$\mathbb{Z} = \\{\dots, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0 , 1 , 2, 3, 4, 5, 6, 7, 8, 9, 10, \dots\\}$$

Power of 10

For example, let’s enter numbers in powers of 10.

Power of 10 Equal Multiple representation of 10
$10^{-1}$ $=$ $\dfrac{1}{10}=0.1$
$10^{-2}$ $=$ $\dfrac{1}{100}=0.01$
$10^{-3}$ $=$ $\dfrac{1}{1000}=0.001$
$10^{-4}$ $=$ $\dfrac{1}{10000}=0.0001$
$10^{-5}$ $=$ $\dfrac{1}{100000}=0.00001$
$10^{-6}$ $=$ $\dfrac{1}{1000000}=0.000001$
$10^{-7}$ $=$ $\dfrac{1}{10000000}=0.0000001$
$10^{-8}$ $=$ $\dfrac{1}{100000000}=0.00000001$
$10^{-9}$ $=$ $\dfrac{1}{1000000000}=0.000000001$
$10^{10}$ $=$ $\dfrac{1}{10000000000}=0.0000000001$
$10^{11}$ $=$ $\dfrac{1}{100000000000}=0.00000000001$
$10^{12}$ $=$ $\dfrac{1}{1000000000000}=0.000000000001$

How do we get the results?

Remember the scale. This way. let’s use this method.

Grafiko's Galileo.
$$ \begin{aligned} 10^{-1} &= \color{red}{10^1} ~.~\dfrac{10^{-1} }{\color{red}{10^1} }\\ &= \dfrac{10^{1-1} }{10 }\\ &= \dfrac{10^{0} }{10 }\\ &= \dfrac{1 }{10 }\\ \end{aligned} $$

Power of 2

With a power concept of 10, we can now talk about the base power 2.

$$ \begin{aligned} 2^{-2} &= \color{red}{2^2} ~.~\dfrac{2^{-2} }{\color{red}{2^2} }\\ &= \dfrac{2^{2-2} }{2^2 }\\ &= \dfrac{1 }{2^2 }\\ &= \dfrac{1 }{4 }\\ \end{aligned} $$
Power of 2 Equal Result
$2^{-1}$ $=$ $\dfrac{1}{2^{1}}=0.5$
$2^{-2}$ $=$ $\dfrac{1}{2^{2}}=0.25$
$2^{-3}$ $=$ $\dfrac{1}{2^{3}}=0.125$
$2^{-4}$ $=$ $\dfrac{1}{2^{4}}=0.0625$
$2^{-5}$ $=$ $\dfrac{1}{2^{5}}=0.03125$
$2^{-6}$ $=$ $\dfrac{1}{2^{6}}=0.015625$
$2^{-7}$ $=$ $\dfrac{1}{2^{7}}=0.0078125$
$2^{-8}$ $=$ $\dfrac{1}{2^{8}}=0.00390625$
$2^{-9}$ $=$ $\dfrac{1}{2^{9}}=0.001953125$
$2^{-10}$ $=$ $\dfrac{1}{2^{10}}=0.0009765625$

Potentiation property

The first property is the multiplication between two powers.

Multiplication of powers

$$\text{base}^m ~.~ \text{base}^n = \text{base}^{m+n}$$

Example: Calculate $2^4 ~.~ 2^5.$

$$2^4 ~.~ 2^5 = 2^{4+5} = 2^9$$

Is the property worth it?

Initially we call the base $a$

$$ \begin{aligned} a^m ~.~a^n &= \underbrace{(a~.~a~.~ \dots ~.~a)}_{m \text{ vezes} }~.~\underbrace{(a~.~a~.~ \dots ~.~a)}_{n \text{ vezes} } = \\ &= \underbrace{(a~.~a~.~ \dots ~.~a)}_{m+n \text{ vezes} }\\ &= a^{m+n} \end{aligned} $$

Now the reciprocal

$$ \begin{aligned} a^{m+n} &= \underbrace{(a~.~a~.~ \dots ~.~a)}_{m+n \text{ vezes} } = \\ &= \underbrace{(a~.~a~.~ \dots ~.~a)}_{m \text{ vezes} }~.~\underbrace{(a~.~a~.~ \dots ~.~a)}_{n \text{ vezes} } \\ &= a^{m} ~.~ a^{n} \end{aligned} $$

The second property is the multiplication between two powers.

Division of powers

$$\dfrac{\text{base}^a }{\text{base}^b} = \text{base}^{a-b}$$

Example: Calculate $\dfrac{10^5}{10^2}$

$$\dfrac{10^5}{10^2} = 10^{5-2} = 10^3$$

Now let’s show this property:

$$ \begin{aligned} \frac{a^m}{a^n} &= \dfrac{\underbrace{\underbrace{(a~.~a~.~ \dots ~.~a)}_{n \text{ vezes} } ~.~ \underbrace{(a~.~a~.~ \dots ~.~a)}_{m-n \text{ vezes} } }_{\color{red}{n} + m - \color{red}{n} ~=~ m} }{\underbrace{(a~.~a~.~ \dots ~.~a)}_{n \text{ vezes} } }\\ &= \dfrac{ \underbrace{(a~.~a~.~ \dots ~.~a)}_{n \text{ vezes} } }{\underbrace{(a~.~a~.~ \dots ~.~a)}_{n \text{ vezes} } }~.~ \underbrace{(a~.~a~.~ \dots ~.~a)}_{m-n \text{ vezes} }\\ &= \underbrace{(a~.~a~.~ \dots ~.~a)}_{m-n \text{ vezes} }\\ &= a^{m-n} \end{aligned} $$

We will leave the converse as an exercise.

It is worth remembering that to use this property, the base must be the same.

Example: Calculate $\dfrac{10^8 ~.~ 3^2}{3 ~.~10^3}$

For the calculation, the property will only be used for powers with the same base. In fact,

$$\dfrac{10^8 ~.~ 3^2}{3 ~.~10^3} = 10^{8-3} ~.~ 3^{2-1} = 10^5 ~.~3 $$

Base inversion in powers

$$\left(\dfrac{\text{base 1} }{\text{base 2} }\right)^{-a} = \left(\dfrac{\text{base 2} }{\text{base 1} }\right)^{a}$$

Example: Show that $\left(\dfrac{3}{2}\right)^{-2} = \left(\dfrac{3}{2}\right)^{2}$

$$ \begin{aligned} \left(\dfrac{3}{2}\right)^{-2} &\overset{\text{balança} }{=} \dfrac{3^{-2} }{2^{-2} } ~.~ \underbrace{\color{red}{\dfrac{3^{2} }{3^{2} } } }_{=1}\\ &= \dfrac{3^{2-2} }{2^{-2}~.~ 3^{2} }\\ &\overset{\text{balança} }{=} \dfrac{1 }{2^{-2}~.~ 3^{2} }~.~ \color{red}{\dfrac{2^{2} }{2^{2} } } \\ &= \dfrac{2^2 }{\underbrace{2^{2-2} }_{\color{red}{2^0=1} }~.~ 3^{2} }\\ &= \dfrac{2^2 }{3^{2} }\\ &= \left(\dfrac{2 }{3 }\right)^2\\ \end{aligned} $$

These concepts are fundamental, because without it we cannot talk about logarithms. Wait for the next posts.