Exponential Arithmetic
Before coming to this blog, I assume you have learned the fundamentals of basic arithmetic (addition, subtraction, multiplication, and division of numbers). $4+4=8$, $10∗6=60$, $7−3=4$, and $\frac{12}{2}=6$ should all be very comfortable to calculate. When it’s $11:37\small{AM}$ and you calculate there are $23$ minutes until lunch at noon you have used arithmetic. When someone hands you a five dollar bill for a $\$4.75$ churro, you hand them $\$0.25$ back and have done arithmetic.
Exponentiating Numbers
Less common in daily experience is exponential arithmetic. We can write two times two as $2*2$, but we can also write it using exponential notation, $2^2$. This is called squaring a number. In this example, we call this, "two squared".
We can square any number we desire:
$$3*3 = 3^2,\ 6*6 = 6^2$$
When we multiply three copies of a number, it is called cubing:
$$3*3*3 = 3^3,\ 6*6*6= 6^3$$
In general, we can multiply as many copies of a number that we want, this is called exponentiation. A base number $b$ is exponentiated by raising it to some power $p$:
$$b^p$$
For the exponentiated number below, $4$ is the base, $8$ is the power:
$$4^8$$
A number alone is multiplied by itself once and is the same as exponentiating to the power of $1$:
$$3 = 3 ^1,\ 6 = 6^1$$
Any number can be exponentiated, even numbers that contain a decimal:
$$2.6*2.6 = 2.6^2$$
Multiplying Exponentials
Suppose we multiply a number $b$ by itself $4$ times:
$$b * b * b * b$$
We can simplify this equation into a single exponential or two multiplied exponentials:
$$b * b * b * b = (b * b) * (b * b) = b^2 * b^2 = b^4$$
From this example, we observe a special property of exponents. If we multiply two exponentiated numbers with equal bases, $b ^p$ and $b ^q$, their exponents are additive:
$$\boxed{b ^p * b ^ q = b ^ {p + q}}$$
When an exponent is negative, the exponentiated number is a fraction:
$$b ^{-1} = \frac{1}{b},\ b ^ {-2} = \frac{1}{b ^2}$$
Multiplying negative and positive exponentiated numbers is also additive:
$$\frac{b * b }{b} = \frac{b ^ 2 } { b ^ 1 } = b ^ 2*b ^{-1} = b ^ {2 - 1} = b ^ 1$$
This additive property of exponents brings us to the inevitable conclusion that exponentiating a number to the zeroth power yields $1$:
$$1 = \frac{b}{b} = b^1* b^{-1} = b ^ {1 - 1}= b^0 = 1$$
Exponentiating Exponentials
Consider an example of an exponential raised to a power, a nested exponential:
$${(2^2)}^2 = (2 * 2)^ 2 = (2 * 2)*(2 * 2) = 2^4$$
This example demonstrates that nested exponentials are multiplicative. An exponential raised to a power results in a new exponential with power equal to the product of the exponents:
$$\boxed{{(b ^ p)} ^ q = b ^ {p*q}}$$
Fractional Exponentials and Roots
A number is squared by raising it to the second power. The act of squaring a number can be undone by raising it to the $\frac{1}{2}$ power so that the exponents cancel:
$${(b ^ 2)}^{\frac{1}{2}} = b ^ {(2)*(\frac{1}{2})} = b$$
This is also called the square root. Two different methods are typically used to symbolize the square root:
$$b^{\frac{1}{2}},\ \sqrt{b}$$
The square root is not the only possible root. It is possible to take the cubed root $b^\frac{1}{3}$ or any order root $b ^{\frac{1}{p}}$:
$$b^{\frac{1}{p}} = \sqrt[p]{b}$$
Consider an example where $b = 64$, and $p = \frac{1}{6}$:
$$64 ^{\frac{1}{6}} = 2 =\sqrt[6]{64}$$
For proof, acknowledge that $2 * 2 * 2 * 2 * 2 * 2 = 2 ^6 = 64$.
Like the square and and square root, a sixth order root cancels a sixth order power:
$$64 ^{\frac{1}{6}} = (2 ^6) ^{\frac{1}{6}} = 2 ^ {(6*\frac{1}{6})} = 2$$
Since fractional exponents are valid, decimal exponents are also valid:
$$\sqrt[3]{b} = b ^\frac{1}{3} = b ^{0.3333333}$$
An exponential like $3^{2.25} = 11.8447$ is perfectly reasonable. Though modern calculators permit us to easily calculate numbers like $3^{2.25}$, calculating the value of this number was at one point very difficult.
Exponentiation is distributive. For two bases $b$ and $h$ multiplied together and then raised to a power $p$, the power distributes:
$$\boxed{(b*h) ^p = b ^p * h ^p}$$
For a demonstration, consider an example where $b = 2$, $h = 3$, and $p = 2$:
$$(2*3) ^ 2 = 2 ^ 2 * 3 ^ 2$$
$$(6) ^ 2 = 4 * 9$$
$$36 = 36$$
Multiplying Exponentials of Dissimilar Bases
When we multiply two exponentials with differing bases $b$ and $h$, they do not combine or simplify in any way unless we can manipulate them to force the bases equal.
For example, consider the case of multiplying the exponentials $4 ^ 3$ and $2 ^ 7$:
$$4 ^ 3 * 2 ^ 7 = ?$$
To simplify this expression, we must use the fact that $4 = 2 ^ 2$. Substituting, we obtain:
$${(2 ^ 2)} ^ 3 * 2 ^ 7 $$
$$\downarrow (Power\:of\:Power\:Rule) \downarrow$$
$$2 ^ 6 * 2 ^ 7 $$
$$\downarrow (Product\:of\:Exponentials\:Rule) \downarrow$$
$$4 ^ 3 * 2 ^ 7 = 2 ^ {13}$$
Summary of Exponential Properties
The principles of exponential arithmetic are very useful. Because of this, they are summarized below.
Definition of Exponentials
An exponential is a number $b$ multiplied by itself $p$ times, symbolized as $b^p$. The number $b$ is called the base while the number $p$ is called the power. Valid examples include $3 ^ 2$, $2 ^{4.2}$, and $25 ^ \frac{1}{2}$.
Product of Exponentials Rule
When we multiply two exponentials with equal base $b$ and differing powers $p$ and $q$, the result is an exponential with base $b$ and power equal to the sum of the powers:
$$b ^ p * b ^ q = b ^ {p + q}$$
Power of Power Rule
When we raise an exponential with base $b$ and power $p$ to another power $q$, the result is an exponential with base $b$ and power equal to multiplication of the powers:
$${(b ^ p)} ^ q = b ^ {p * q}$$
Distributing Exponentials
Exponentiation is distributive. If we have two numbers $b$ and $h$, multiplied together and raised to a power $p$, the power is distributed:
$$({b*h})^p = b ^ p * h ^ p$$