2.1 Contents

1. Universal Properties

2. Negative Exponents

3. Power Rule

4. Higher Exponentials

5. Power Distribution

6. Roots

7. Rational Exponents

8. Root Expansion & Factoring

9. Even Roots of Even Powers

1. Scientific Notation

2.1.1 Universal Properties

Exponentiation is repeated multiplication. An exponent is the number of times to "times" a number.

• Exponents do not commute: $x^a\ne a^x$
• Exponents do not associate: $x^{(a^b)}\ne {(x^a)}^b$
• Factoring is the condensing of terms: $x^3$
• The reverse of factoring is expansion: $x\cdot x\cdot x$

2.1.2 Negative Exponents

$$x^{-a}=\frac{1}{x^a}$$

Deductive Logic

Multiplication is reversible with division, therefore exponential reduction yields inverses once reaching negatives.

$$x^3=1\cdot x\cdot x\cdot x$$	$$3^3=27$$
$$x^2=1\cdot x\cdot x$$	$$3^2=9$$
$$x=1\cdot x$$	$$3^1=3$$
$$x^0=1$$	$$3^0=1$$
$$x^{-1}=1/x$$	$$3^{-1}=1/3$$
$$x^{-2}=1/(x\cdot x)$$	$$3^{-2}=1/9$$
$$x^{-3}=1/(x\cdot x\cdot x)$$	$$3^{-3}=1/27$$

2.1.3 Power Rule

$$x^a\cdot x^{\pm b}=x^{a \pm b}$$

Deductive Logic

$x$ is multiplied $a$ times and $x$ is multiplied $b$ times, and if the terms are multiplied to each other, this is a partially expanded form

Example

$$2^3\cdot 2^4=8\cdot 16=128=2^7$$ $$2^3\cdot 2^4=2^{3+4}=2^7=128$$

2.1.4 Higher Exponentials

Evaluate starting from the highest exponent first, working downward, minding parenthesized terms along the way. $$x^{a^{b^{c^d}}}=x^{\Big(a^{\big(b^{(c^d)}\big)}\Big)}$$

Tetration

A unique case of one value in repeated exponentiation, often represented as $$x^{x^{x^{x^x}}}=\phantom{}^5x$$

2.1.5 Power Distribution

$${(x^a\cdot y^{\pm b})}^c=x^{a\cdot c}\cdot y^{\pm b\cdot c}$$

Deductive Logic

Distribute the exponential $c$ by expanding into ${(x^a)}^c\cdot {(y^{±b})}^c$. The terms are multiplied $c$ times

Example 1

$$(2^3)^2=2^{(3\cdot 2)}$$ $$8^2=2^6=64$$

Example 2

Given $3\cdot x–5\cdot y=2$, evaluate $8^x/32^y$ $$\frac{8^x}{32^y}=\frac{{(2^3)}^x}{{(2^5)}^y}=\frac{2^{3\cdot x}}{2^{5\cdot y}}$$ Apply the power rule, then substitute $3\cdot x–5\cdot y$ for $2$ $$\frac{2^{3\cdot x}}{2^{5\cdot y}}=2^{3\cdot x-5\cdot y}=2^2$$

2.1.6 Roots

Reciprocal exponents represent roots and are multiplicative inverses of their integer counterparts. $$x^{1/n}=\sqrt[n]{x}$$

2.1.7 Rational Exponents

$$x^{p/q}={(\sqrt[q]{x})}^p=\sqrt[q]{x^p},\medspace\forall x≥0$$

2.1.8 Root Expansion & Factoring

$$\sqrt[n]{x\cdot y^{\pm 1}}=\sqrt[n]{x}\cdot \sqrt[n]{y^{\pm 1}},\medspace x>0 \land y>0$$

Rule

Even though $1^2=1$ and $(-1)^2=1$, radicands cannot be factored into two negatives because it creates the following logical fallacy

$\qquad 2=2$
$\qquad 2=1+1$
$\qquad 2=1+\sqrt{1}$
$\qquad 2=1+\sqrt{-1\cdot -1}$
$\qquad 2=1+\sqrt{-1}\cdot \sqrt{-1}$
$\qquad 2=1+(\sqrt{-1})^2$
$\qquad 2=1-1$
$\qquad 2=0$
$\qquad $No solution

2.1.9 Even Roots of Even Powers

$$\sqrt[n]{x^n}=|x|,\medspace\{ 2\cdot n | n \isin \Z \}$$

2.1.A Scientific Notation

All numbers are multiples of ten to a power in accordance with the leading digit $$299792458=2.99792458\cdot 10^8$$ $$0.0820574=8.20574\cdot 10^{-2}$$