Exponentiation

You are probably already somewhat familiar with exponentiation operations; they – like subtraction, multiplication, and division – are an extension of addition. Herein, we will cover the basics of exponentiation as well as some of the fundamental laws and principles concerning these operations.

0. Summary of the Laws of Exponentiation

$x^a\cdot x^b = x^{a+b}$
$\frac{x^a}{x^b} = x^{a-b}$
$(x^a)^b = x^{ab}$
$\left(x^ay^b\right)^c = x^{ac}y^{bc}$
$\left(\frac{x^a}{y^b}\right)^c = \frac{x^{ac}}{y^{bc}}$

1. Exponentiation: The “Multiplication of Multiplications”

You might have thought arithmetic stopped there? However, there are many more operations that you have likely used before that are also considered in the realm of arithmetic: exponentiation – including powers, roots, and logs – as well as factorialization.

Like multiplication is simply an operation that expresses a sequence of successive additions, exponentiation is an operation that expresses a sequence of successive multiplications. In this way, exponentiation is a further extension of addition; it is a repeated addition of many similar additions!

$$3^3=(3)(3)(3)=\>\text{by the associative property}\>$$

$$\bigg[(3)(3)\bigg](3)=(3+3+3)(3)=(3+3+3)+(3+3+3)+(3+3+3)=27$$

In this way, exponentiation is merely a compact way of writing a particular repeated pattern of addition operations derived from the base number and a part called the exponent. We are interested in this particular pattern because it turns out to be useful in quantifying obersarvations about the universe around us.

Exponents – often known as ‘powers’ – are numbers that indicate how many times a base number may be multiplied by itself. This operation can be written in two standard ways:

$2^3$ (read as “two to the third power”)

2^3 (read the same way)

This second method of writing exponentiation (i.e. that with a caret sign) is used in typing when superscripting (as in the first method) is unavailable or cumbersome.

For instance, the number $4^3$ instructs you to multiply four by itself three times. The base is the number being raised by a power, whereas the exponent (or power) is the superscript number (or expression) above it (or the number/expression after the carrot sign).

It is worth noting that the power of two is also known as “squared” and the power of three is known as “cubed.” Sometimes, though rarely, the fourth power is called “hypercubed.” All other powers have no special designations.

The base and/or exponent in exponentiation can also be “expressions” or “operations” themselves and not only numbers. Consider some of the examples to the right, which contain more and less complex examples of exponentiation expressions.

Examples of Exponentiation Expressions:

$2^x\quad$ (read as “two to the power of x”)

$4^{3^2}\quad$ (read as “four to the power of three squared”)

$x^2\quad$ (read as “x to the power of 2” or “x squared”)

$3^{(7x-4)}\quad$ (read as “three to the power of the quantity 7x minus 4”)

$(5-\tfrac{4}{3})^{(3-\tfrac{1}{2})}\quad$ (read as “the quantity 5 minus 4/3s to the power of the quantity 3 minus one half”)

$(x-y)^a\quad$ (read as “the quantity of x minus y to the power of a”)

Zero Power Rule:

Any base that has been raised to the power of zero equals one. No matter what you raise to the power of zero is 1.

$$3^0=1\quad9^0=1\quad0^0=1\quad10,000,000^0=1\quad{x^0=1}\quad\&^0=1\quad$$

Note that $0^0=1$ is defined by convention due to principle of “continuity” on the fuction $x^x$, which we will examine in a later unit.

Negative Exponent Rule:

When a base is raised to a negative exponent, this tells us to take the reciprocal of the exponentiation expression.

To reciprocate a number, use the following formula:

Make a fraction out of the number (put it over one)
Change the denominator to the numerator and vice versa (i.e. flip the fraction).

$$2^{-2}=\frac{1}{2^2}=\frac{1}{4}\quad(\frac{3}{7})^{-1}=\frac{1}{\frac{3}{7}}=$$

$$\frac{7}{3}\quad b^{-3}=\frac{1}{b^3}\quad {x^{-a}=\frac{1}{x^a}}\quad(x+y)^{-2}=\frac{1}{(x+y)^2}$$

Remember that the exponent only effects the thing it is directly appended to. So, if we are also multiplying the base by something else, that something else (constant, variable, etc.) is not exponentiated in any way. For example:

$$5(2^{-2})=5(\frac{1}{2^2})=5(\frac{1}{4})=\frac{5}{4}$$

$$4b^{-3}=4(\frac{1}{b^3})=\frac{4}{b^3}$

2. The Five Laws of Exponents:

You must understand seven exponent rules, often known as exponent laws. Each rule demonstrates how to answer various sorts of arithmetic problems as well as how to multiply, divide, and add exponents.

Product of powers rule
Quotient of powers rule
Power of a power rule
Power of a product rule
Power of a quotient rule

2.1 Product of Powers Rule:

If we are given exponentiations with the same base multiplied together (product), we can simplify the expression by raising the base to the sum of the exponents.

$$5^3(5^2)=5^{3+2}=5^5$$

$$11^5(11^{-2})=11^{5+-2}=11^3$$

$$a^7(a^3)=a^{7+3}=a^10$$

Why does this work? Well, let’s take the first example from above and remember that all exponentiations are sequential multiplication operations.

$$5^3(5^2)=(5\times5\times5)(5\times5)$$

And, by the Associative Property of Multiplication, we know that the grouping of these fives does not matter. So, we can simply group them all together.

$$5^3(5^2)=(5\times5\times5)(5\times5)=5\times5\times5\times5\times5=5^5$$

A more general example could be:

$$n^4(n^5)=(n)(n)(n)(n)\times(n)(n)(n)(n)(n)=n(n)(n)(n)(n)(n)(n)(n)(n)=n^9$$

Generally stated, the Product of Powers Rule is:

$$x^y(x^z)=x^{y+z}$$

2.2 Quotient of Powers Rule:

If we are given exponentiations with the same base , one divided by the other (quotient), we can simplify the expression by raising the base to the difference of the exponents.

$$\frac{5^3}{5^2}=5^{3-2}=5^1=5$$

$$\frac{11^5}{11^{-2}}=11^{5-(-2)}=11^{5+2}=11^7$$

$$\frac{a^7}{a^3}=a^{7-3}=a^4$$

Why does this work? Well, let’s take the first example from above and remember that all exponentiations are sequential multiplication operations.

$$\frac{11^5}{11^-2}=\frac{11(11)(11)(11)(11)}{\frac{1}{11^2}}=$$

$$\left[11(11)(11)(11)(11)\right](11^2)=\left[11(11)(11)(11)(11)\right](11)(11)=$$

$$(11)(11)(11)(11)(11)(11)(11)=11^7$$

$$\frac{5^3}{5^2}=\frac{5\times5\times5}{5\times5}=\frac{5}{5}\times\frac{5}{5}\times\frac{5}{1}=(1)(1)(5)=5^1$$

A more general example could be:

$$\frac{x^4}{x^5}=\frac{x(x)(x)(x)}{x(x)(x)(x)(x)}=$$

$$\frac{x}{x}\left(\frac{x}{x}\right)\left(\frac{x}{x}\right)\left(\frac{x}{x}\right)\left(\frac{1}{x}\right)=$$

$$(1)(1)(1)(1)\left(\frac{1}{x}\right)=\frac{1}{x}=x^{-1}$$

Generally stated, the Product of Powers Rule is:

$$\frac{x^y}{x^z}=x^{y-z}$$

Proof: Zero Power Rule

Since we proved the Quotient of Powers Rule from first principles (namely addition via multiplication as shown directly above, we can use it to prove the Zero Power Rule (i.e. $a^0=1$, which might not be obvious from first principles.

Let $c=a^b$ for some numbers $a$, $b$, and $c$. Then.

$$1=\frac{c}{c}=\frac{a^b}{a^b}=a^{b-b}=a^0$$

Thus, an number, call it $a$, to the zeroth power is equal to 1. This is:

$$a^0=1$$

2.3 Power of a Power Rule:

We can also take powers sequentially. If we are given a single base raised to two or more power successively, then we can simply raise that same base to the product of the exponents.

$$(5^2)^3=5^{2(3)}=5^6$$

$$(11^5)^{-2}=11^{5(-2)}=11^{-10}$$

$$(a^7)^3=a^{7(3)}=a^{21}]$$

Why does this work? Well, let’s take the first example from above and remember that all exponentiations are sequential multiplication operations.

$$(5^2)^3=(5\times5)(5\times5)(5\times5)$$

And, by the Associative Property of Multiplication, we know that the grouping of these fives does not matter. So, we can simply group them all together rather than in three groups of two.

$$(5^2)^3=(5\times5)(5\times5)(5\times5)=5\times5\times5\times5\times5\times5\times=5^6

A more general example could be:

$$(x^4)^3=\left[x(x)(x)(x)\right]\times\left[x(x)(x)(x)\right]\times\left[x(x)(x)(x)\right]=x(x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x)=x^12$$

Generally stated, the Product of Powers Rule is:

$$(x^y)^z=x^{yz}$$

2.4/5 Power of a Product/Quotient Rule:

We can also distribute an exponent to and from a collection of summed bases, assuming those bases have all have exponents containing a common factor.

$$(2\times5)^2=2^2(5^2)$$

$$(-4\times7)^{11}=(-4)^{11}(7)^{11}$$

$$(xy)^3=x^3y^3$$

If there are exponents linked to the base, this rule also applies.

$$(x^2y^3)^5=x^{2(5)}y^{3(5)}=x^{10}y^{15}$$

Why does this work? Well, let’s take the example below and remember that all exponentiations are sequential multiplication operations.

$$(5^3\times2^4)^2=(5\times5\times5\times2\times2\times2\times2)^2=(5)(5)(5)(2)(2)(2)(2)\times(5)(5)(5)(2)(2)(2)(2)$$

And, by the Associative Property of Multiplication, we know that the grouping of these fives and twos does not matter. So, we can simply group the like numbers all together rather than in three groups of two.

$$(5^3\times2^4)^2=(5)(5)(5)(2)(2)(2)(2)\times(5)(5)(5)(2)(2)(2)(2)=(5)(5)(5)(5)(5)(5)\times(2)(2)(2)(2)(2)(2)(2)(2)=5^6(2^8)$$

A more general example could be:

$$(n^4)^3=(n)(n)(n)(n)\times(n)(n)(n)(n)\times(n)(n)(n)(n)=(n)(n)(n)(n)(n)(n)(n)(n)(n)(n)(n)(n)=n^{12}$$

Generally stated, the Product of Powers Rule is:

$$(xy)^z=x^{z}y^{z}$$

And since multiplication is translatable into division, we could say that

$$\left(\frac{x}{y}\right)=\left(x\cdot\frac{1}{y}\right)^z=(xy^{-1})^z=x^z(y^{-1})^z=x^zy^{-z}=\frac{x^z}{y^z}$$

Thus, the Power of a Quotient Rule is:

$$\left(\frac{x}{y}\right)=\frac{x^z}{y^z}$$