2.1 Contents

• Universal Properties
• Negative Exponents
• Power Rule
• Higher Exponentials
• Power Distribution
• Roots
• Rational Exponents
• Root Expansion & Factoring
• Even Roots of Even Powers
• 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: $x^a≠a^x$
• Exponents do not associate: $x^{(a^b)}≠{(x^a)}^b$
• Factoring is the condensing of terms: $x^3$
• The reverse of factoring is expansion: $x·x·x$

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.

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

2.1.3 Power Rule

$$x^a·x^{\pm b}=x^{a \pm b}$$

Deductive Logic

With the same base, $x^a$ is $x$ multiplied $a$ times, $x^b$ is $x$ multiplied $b$ times, and they are multiplied to each other, so then the exponents are added $$2^3·2^{-5}=2^{3-5}=2^{-2}=1/4$$

2.1.4 Higher Exponentials

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

2.1.5 Power Distribution

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

Deductive Logic

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

Example

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

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

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

Rule

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

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

2.1.9 Even Roots of Even Powers

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

2.1.A Scientific Notation

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