$\color{#FFF}\text{"The greatest shortcoming of the human race}$ $\color{#FFF}\text{is our inability to understand }$ $\color{#FFF}\text{the exponential function."}$

$\color{#FFF}\text{Albert A. Bartlett}$

🏵 Exponents

The Magic of Scaling

🦠 Exponentiation governs phenomena ranging from population growth to how fast computers process data. It can be viewed as repeated multiplication, but it also has profound implications for large-scale and very small-scale processes. Let’s explore what exponents are, how they work, and why they are so powerful.

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🧫 Imagine microbia in a Petri dish. If $( x )$ represents the amount of germs inside, then raising it to the power $( n )$ means every one germ creates $n$ new germs. It scales its contents by its own size, over and over.

⚗️ This is how exponentiation leads to rapid growth, that it's proportional to its own size. Exponents also models self-proportional growth.

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    A-->B --> B2
    A-->C --> C2
    A-->D --> D2
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    classDef Petry2 fill:#044,stroke:#4FF,stroke-width:2px;
    class A,B,C,D Petry;
    class B2,C2,D2 Petry2;

🔰 What is an Exponent?

⚜️ At its simplest, an exponent is calculated with a repeated multiplication. An EXPONENT works the same as the number of times the BASE NUMBER is multiplied by itself.

When we say (x^n), we mean:

\color{#FF3303}
x^n = \underbrace{ x·x...x} _{n-times}

• Base ((x)): The number being multiplied repeatedly.
• Exponent ((n)): The number of times the base multiplies itself.

⚜️ When talking $\color{red}natural$ exponents, think of it like a multiplier on repeat, the base number just keeps getting stacked!

⚜️ This is the most intuitive definition. However, this alone doesn’t explain what happens when exponents are fractional, irrational, or even complex.

\color{#FF0F80}4^3 = 4 ⨯ 4 ⨯ 4 = 64

We read (4^3) as “four to the third power.”

📕 4 is the base number. The raised small number 3 to the right of the base number indicates the number of times the base number is multiplied by itself.

🔍 Invisible Exponents

If a base doesn’t show an exponent, it secretly has an “invisible” exponent of 1:
$\color{#E9190F} x = x^1$

Any base raised to the power of 0 equals 1:
$\color{#FE4E00} x^0 = 1$

🔍 Handling Negative Numbers with Exponents

Be careful when calculating negative numbers with exponents. Parentheses can make all the difference.

$\color{#FFAE03}-3^2 = −(3^2) = −(3 ⨯ 3) = −9$
VS.
$\color{#FFAE03}(−3)^2 = (−3) ⨯ (−3) = 9$

In the first example, only 3 is raised to the second power, but in the second example, -3 is raised as a whole due to the parentheses.

🔍 Simplifying Expressions with Exponents

When combining exponents, the base must be the same. Here are the key rules:

Multiplying Exponents:

• $\color{#FE4E00}x^a ⨯ x^b = x^{a+b}$
• $\color{#FE4E00}x^a x^b = x ^{(a+b)}$

\color{#E67F0D}
a^n · a^m = \underbrace{ a·a...a} _{n-factores} ⨯ \underbrace{ a·a...a} _{m-factores} = \underbrace{ a·a...a} _{n+m-factores} = a^{n+m}

Dividing Exponents:

• $\color{#FE4E00} x^a \div x^b = x^{a-b}$
• $\color{#FE4E00}x^a ÷ x^b = x^{(a-b)}$
• $\color{#FE4E00}\frac{x^a}{x^b} = x^{(a-b)}$

Power of a Power:

When there is an exponent inside parentheses and another outside the parentheses, this is called a POWER OF A POWER. A power of a power can be simplified by multiplying the exponents. It looks like this:

• $\color{#E9190F}{(v^a )}^b = v^{(a ⨯ b)}$

🔍 Negative Exponents

A negative exponent simply flips the number into a reciprocal. Think of it like flipping the fraction upside down.

\color{#FF0F80}x^{-m} = \frac{1}{x^{m}} = \ddot{x}^{m}

• Move a negative exponent from numerator to denominator: $\color{#FF0F80} x^{-m} = \frac{1}{x^m}$

• Or move it from denominator to numerator: $\color{#FF0F80} \frac{1}{x^{-m}} = x^m$

💡 Distributing Exponents over Products and Quotients

We find very useful that exponentiation is distributive over multiplication.

\color{#FF1F91}
   (ab)^n = a^n \, b^n \quad\quad\text{and}\quad\quad
   \left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}.

🏵 Scientific Notation

⚜️ When numbers get very big or very small, we use scientific notation to keeps them manageable. They are expressed as a product of a number between 1 and 10 and a power of 10. This is, we use a small number followed by the Order of the quantity. This means: what is the magnitude of ten we are working over? Thousands? Millions? Billions?

\color{#E67F1E}
aEⅩb ⟺ a·⏨b

\color{#E67F1E}
\text{with  } a∈[1,10), b∈ℤ

For example

\begin{matrix}
1EⅩ1 = 1 \\
2EⅩ3 = 20 \\
3EⅩ4 = 3000 \\
5EⅩ-6 = 0.000005 \\
\end{matrix}

3·10^9 = 3.000.000.000

🏵 Radicals

Radicals, or roots, provide the inverse operation to exponentiation.

For this purpose, we want consistency! If $(x^2)$ means “multiply $(x)$ by itself twice,” then $(x^{\tfrac12})$ should represent the operation that “undoes” squaring—namely, taking the square root. By extension, $(x^{\tfrac{m}{n}})$ represents “the $(n)$-th root of $(x^m)$.”

• The nth root of $a$ is the number $b$ such that $b^n = a$.

  \color{#FE4E00}
  \sqrt[n]{a} = b \quad \text{ ⟺ } \, b^n = a

• Radicals can also be written as fractional exponents:

  \color{#E9190F}
  \sqrt[n]{a} = a^{1/n} = a^\ddot{n}

🏵 Exponents' Rules

\color{#FF0F80}
\text{Product Rule: }
 x^{m} \cdot x^{n}=x^{m+n}

\color{#FFAE03}
\text{Quotient Rule: }
\frac{x^{m}}{x^{n}} =x^{m-n}

\color{#E67F0D}
\text{Power of a Power: }
{(x^{m})}^{n} =x^{m\cdot n}

\color{#FE4E00}
\text{Negative Exponents: }
{x}^{-n} = ẍ^{n} = \frac{1}{{x}^{n}}

\color{#E9190F}
\text{Distributing Over Multiplication: }
{(xy)}^{n} = x^{n}y^{n}

\color{#FF0F80}
\text{Distributing Over Division: }
{(\frac{x}{y})}^{n} = \frac {x^{n}}{y^{n}}

\color{#E9190F}
\text{Radicals: }
\sqrt[n]{x} = x^{\frac{1}{n}}

📈 Visualizing Exponential Growth

Exponential growth can be visualized with a quick chart. Here's what happens when you square numbers from 1 to 15:

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    title "Exponential Growth"
    x-axis "x" [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ]
 y-axis "y" 0 --> 225
    line [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 ]

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Interpreting Exponential Growth

Now, what happens when we use a fixed number raised to a variable exponent?

For truly explosive growth, consider (2^x), where doubling at each step leads to dramatically higher numbers in short order.

$$ 2^x $$

This means the number 2 multiplies itself x times.

Real-World Interpretation

Exponentiation like $( 2^x )$ is everywhere in nature and technology:

• Doubling Growth: If something doubles every unit of $( x )$, how much will exist after $( x )$ steps?
  • Example: If a bacteria population doubles every hour, the number of bacteria after $( x )$ hours is $( 2^x )$.
• Computing Power: The speed of computers often follows exponential trends. The number of transistors in computer chips has doubled approximately every 18 months (Moore's Law), following a pattern like:

  N_{\text{transistors}} \approx 2^t

where $( t )$ is time.

• Sound and Decibels: The way we measure sound intensity follows an exponential scale. If one sound is 10 times stronger than another, it means it has an exponent of 10 in its intensity.

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Special Cases of Exponents

Here are some important exponentiation rules:

Rule	Explanation	Example
$( x^0 = 1 )$	Any number raised to the power of 0 is 1.	$( 5^0 = 1 )$
$( x^1 = x )$	Any number raised to the power of 1 is itself.	$( 7^1 = 7 )$
$( x^{-n} = \frac{1}{x^n} )$	A negative exponent means "take the reciprocal."	$( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} )$
$( x^{\frac{1}{n}} = \sqrt[n]{x} )$	A fractional exponent represents a root.	$( 9^{\frac{1}{2}} = \sqrt{9} = 3 )$

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Conclusion

Exponents are multiplication on steroids. They allow numbers to grow (or shrink) at an astonishing rate, which is why they're used to model natural growth, physics, finance, and even digital data processing.