Integrated Math 1 · Chapter 1 · Patterns, Functions and the Algebra Behind Them

Working with Exponents

Every rule in this section is derived by counting factors, never announced. Expand, count, collapse — and you can rebuild any of them from scratch.

bookSHelf · Integrated Math 1 · §1.5 · a self-paced section

Nine properties, one method: write the factors out and count them.

Working with Exponents · bookSHelf Integrated Math 1§1.5

Outline — by the end of this section you will be able to

Objectives

Read an exponential expression correctly, and tell $-5^2$ from $(-5)^2$ Def 1.5.1
Derive and use the product property by counting factors Def 1.5.2
Derive and use the quotient property by dividing out common factors Def 1.5.3
Explain why $a^0 = 1$ and why $a^{-n} = \dfrac{1}{a^n}$ , rather than memorizing them Def 1.5.4–1.5.6
Use the power properties on a power, a product and a quotient Def 1.5.7–1.5.8
Simplify with several properties at once, leaving only positive exponents Ex 1.5.5–1.5.6
Convert both directions between decimal form and scientific notation Def 1.5.9

The seven section SLOs verbatim.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.1 — What an exponent records

The exponent is a count, and parentheses decide what is counted

Expression	Base	Meaning	Value
$(-5)^4$	$-5$	$(-5)(-5)(-5)(-5)$	$625$
$-5^4$	$5$	$-(5 \cdot 5 \cdot 5 \cdot 5)$	$-625$
$(-2)^6$	$-2$	six factors of $-2$	$64$
$-2^6$	$2$	$-(2^6)$	$-64$

Table 1.5.1: the first place people lose points, and it is notation rather than mathematics.

Parentheses decide the base. Whatever sits immediately inside them is what gets multiplied by itself; otherwise the exponent is evaluated before the negation. When in doubt, expand — if you can write the factors out, you can count them.

The same question — what exactly is the base? — returns in §1.5.5 to separate 5y^-1 from (5y)^-1.

Working with Exponents · bookSHelf Integrated Math 1§1.5

Insight — an exponent is a tally, not an operation

The raised number is not something you do

It is a count of how many copies of the base are being multiplied, the way a receipt says "×3" beside an item.

Read $a^m$ as "there are $m$ of these, all multiplied together," and every rule in this section turns into counting.

The framing the whole section runs on.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.1 — Definition

Exponential Notation

Definition 1.5.1 — Exponential Notation, Base, and Exponent

For a real number $a$ and a counting number $m$ ,

a^m = \underbrace{a \cdot a \cdot \ \cdots \ \cdot a}_{m \text{ factors}}

The number $a$ is the base; $m$ is the exponent, and it tells how many times the base is used as a factor.

An exponent of 1 is usually invisible — $a$ means $a^1$ , and several properties below need it written explicitly first.

Definition 1.5.1: the exponent counts the factors, and the base is the thing being counted.

4^3 = 4·4·4 = 64; 15^1 = 15.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.1 — Try It Now

Try It Now 1.5.1

Try It Now 1.5.1 — the base is whatever the parentheses hold

Evaluate $\;\left(-\dfrac{1}{4}\right)^4\;$ and $\;-\left(\dfrac{1}{4}\right)^4$ , and write $\;(0.63)^2\;$ as a product before evaluating it.

\left(-\dfrac{1}{4}\right)^4 = \dfrac{1}{256} \qquad -\left(\dfrac{1}{4}\right)^4 = -\dfrac{1}{256} \qquad (0.63)^2 = (0.63)(0.63) = 0.3969

Four negative factors pair off into positives; in the second the minus sign waits outside.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.2 — The product property

Nothing was decided; the count decided it

What could $x^2 \cdot x^3$ possibly be? Expand both factors and stop thinking about exponents entirely.

x^2 \cdot x^3 = \underbrace{(x \cdot x)}_{2 \text{ factors}} \cdot \underbrace{(x \cdot x \cdot x)}_{3 \text{ factors}} = \underbrace{x \cdot x \cdot x \cdot x \cdot x}_{5 \text{ factors}} = x^5

And $5$ is $2 + 3$ , which is not a coincidence: the exponents were the counts, and the counts got pooled.

Two factors, then three more, all multiplied together — five factors total.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.2 — Definition

Product Property for Exponents

Definition 1.5.2 — Product Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

a^m \cdot a^n = a^{m+n}

To multiply powers with the same base, keep the base and add the exponents.

The base can be a letter, a number, or a whole parenthesized quantity: $(2b)^{10} \cdot (2b)^{3} = (2b)^{13}$ . Only sameness matters.

Definition 1.5.2: pooling two groups of identical factors adds their counts.

Check: 2^2 · 2^3 = 4 · 8 = 32 = 2^5.

Working with Exponents · bookSHelf Integrated Math 1§1.5

Context Pause — two different jobs in one line

Coefficients multiply; exponents add

3x^4 \cdot 5x^2 = 3 \cdot 5 \cdot x^4 \cdot x^2 = 15 \cdot x^{4+2} = 15x^6

The coefficients get multiplied — $3 \cdot 5 = 15$ — because they are ordinary factors. The exponents get added — $4 + 2 = 6$ — because they are counts of factors, and counts get pooled.

Swapping those two jobs is the error this subsection exists to prevent: $8x^6$ added the coefficients, and $15x^8$ multiplied the exponents. Neither is $15x^6$ .

The single most common error in the whole section.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.2 — Worked example

Naming the reason behind each step

Example 1.5.1 — Coefficients multiply while exponents add

Simplify $\;(3x^2)(-4x^3)$ , naming the reason behind each step.

1 — Regroup (commutative property): $\;3 \cdot (-4) \cdot x^2 \cdot x^3$

2 — Two separate jobs: $3 \cdot (-4) = -12$ , and the exponents pool as $2 + 3$ .

-12x^5

Answer: -12x^5.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.2 — Try It Now

Try It Now 1.5.2

Try It Now 1.5.2 — and one that is not a product at all

Simplify $\;b^5 \cdot b^9$ , then $\;(2m^3)(7m^4)$ , and finally explain why $\;m^3 + m^3\;$ is not $m^6$ .

b^5 \cdot b^9 = b^{14} \qquad (2m^3)(7m^4) = 14m^7 \qquad m^3 + m^3 = 2m^3

That plus sign means addition, so no factors are being pooled — one batch of $m^3$ plus another batch is two batches. The product property applies to products only.

Part 3 is the point: the property covers products and not sums.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.3 — The quotient property

Every factor downstairs takes one upstairs off the floor

Each downstairs $x$ pairs with an upstairs $x$ , and each pair is $\dfrac{x}{x} = 1$ . Whatever is left over stays on the side that had more of it.

\dfrac{x^5}{x^2} = \dfrac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} = x^3 \qquad\qquad \dfrac{x^2}{x^5} = \dfrac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x} = \dfrac{1}{x^3}

And $3 = 5 - 2$ both times. The subtraction is just bookkeeping on how many factors were left over — and dividing a quantity out entirely leaves 1, not nothing.

The disguised-1 move from §0.5.2, run backwards.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.3 — Definition

Quotient Property for Exponents

Definition 1.5.3 — Quotient Property for Exponents

If $a \neq 0$ and $m$ and $n$ are integers, then

\dfrac{a^m}{a^n} = a^{m-n} \ \ (m > n), \qquad \dfrac{a^m}{a^n} = \dfrac{1}{a^{n-m}} \ \ (n > m)

Keep the base and subtract the exponents. The condition $a \neq 0$ is not decoration — §0.5.4 established that division by zero has no value.

Coefficients behave as they did in §1.5.2: $\dfrac{12x^7}{3x^2} = \dfrac{12}{3} \cdot \dfrac{x^7}{x^2} = 4x^5$ .

Definition 1.5.3: pairing off common factors leaves the difference of the counts.

Two forms of one property is awkward — §1.5.5 absorbs the second into the first.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.3 — Worked example

Split the number part from the variable part

Example 1.5.2 — Coefficients divide while exponents subtract

Simplify $\;\dfrac{12x^7}{3x^2}\;$ and $\;\dfrac{b^4}{b^9}$ , writing each answer without a negative exponent.

\dfrac{12x^7}{3x^2} = \dfrac{12}{3} \cdot \dfrac{x^7}{x^2} = 4x^5

Four $b$ 's upstairs pair off with four of the nine downstairs, leaving $9 - 4 = 5$ below and a 1 above.

\dfrac{b^4}{b^9} = \dfrac{1}{b^{9-4}} = \dfrac{1}{b^5}

Answers: 4x^5 and 1/b^5.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.3 — Try It Now

Try It Now 1.5.3

Try It Now 1.5.3 — both directions, then coefficients too

Simplify $\;\dfrac{n^{11}}{n^4}$ , then $\;\dfrac{n^4}{n^{11}}$ , then $\;\dfrac{15c^8}{5c^3}$ .

\dfrac{n^{11}}{n^4} = n^{7} \qquad \dfrac{n^4}{n^{11}} = \dfrac{1}{n^{7}} \qquad \dfrac{15c^8}{5c^3} = 3c^5

The leftovers stay on whichever side started with more of them.

Answers: n^7; 1/n^7; 3c^5.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.4 — The zero exponent

The value of $3^0$ is not invented; it is forced

Power	Value	Step from above
$3^4$	$81$	—
$3^3$	$27$	$81 \div 3$
$3^2$	$9$	$27 \div 3$
$3^1$	$3$	$9 \div 3$
$3^0$	$1$	$3 \div 3$

Table 1.5.2: every step down divides by 3, because each step removes one factor of 3.

The quotient property said nothing about $m = n$ , and that gap is where something new appears. Two honest routes must agree:

\dfrac{a^m}{a^m} = 1 \qquad\text{and}\qquad \dfrac{a^m}{a^m} = a^{m-m} = a^0

So $a^0 = 1$ — the only value that keeps the quotient property working, which is a much stronger reason than a definition handed down.

Two derivations, one from the property and one from the descending pattern.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.4 — Definition

Zero Exponent Property

Definition 1.5.4 — Zero Exponent Property

If $a$ is a nonzero real number, then

a^0 = 1

Any nonzero number raised to the zero power equals 1: $\;7^0 = 1$ , $\;(-8)^0 = 1$ , $\;(-ab)^0 = 1$ .

The condition $a \neq 0$ excludes $0^0$ , and the table shows why: reaching it means dividing by the base at every step, and dividing by 0 is undefined.

Definition 1.5.4: continuing the divide-by-the-base pattern forces the value 1.

a^0 has no meaning under the original definition — the pattern supplies it.

Working with Exponents · bookSHelf Integrated Math 1§1.5

Insight — walking down a staircase

You do not get to invent a step height

Each step down the table is one step down a staircase, and every step is the same height — divide by 3.

When you reach the step labelled $3^0$ , the staircase tells you where the next step lands, and it lands on 1.

The same staircase keeps going below zero in §1.5.5.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.4 — Try It Now

Try It Now 1.5.4

Try It Now 1.5.4 — what exactly is the base, again

Simplify $\;(-8)^0$ , $\;-8^0$ , and $\;5x^0$ , assuming no base is zero.

(-8)^0 = 1 \qquad -8^0 = -(8^0) = -1 \qquad 5x^0 = 5 \cdot 1 = 5

In the third, the exponent touches only $x$ , so the 5 is untouched.

Answers: 1; -1; 5.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.5 — Negative exponents

The staircase has no reason to stop at $3^0$

Power	Value	Step from above
$3^1$	$3$	$9 \div 3$
$3^0$	$1$	$3 \div 3$
$3^{-1}$	$\dfrac{1}{3}$	$1 \div 3$
$3^{-2}$	$\dfrac{1}{9}$	$\dfrac{1}{3} \div 3$
$3^{-3}$	$\dfrac{1}{27}$	$\dfrac{1}{9} \div 3$

Table 1.5.3: nothing about the step "divide by 3" ever runs out.

A negative exponent produces the reciprocal of the matching positive power. The quotient property confirms it — take $\dfrac{x^2}{x^5}$ and refuse to case-split:

\dfrac{x^2}{x^5} = x^{2-5} = x^{-3}

But §1.5.3 already cancelled that same quotient to $\dfrac{1}{x^3}$ . One expression, two correct computations, so $x^{-3} = \dfrac{1}{x^3}$ .

Again nothing was chosen — the pattern had one continuation.

Working with Exponents · bookSHelf Integrated Math 1§1.5

Insight — the minus sign is a floor number, not a temperature

It says which floor, never which side of zero

A negative exponent tells you which floor of the fraction the factor lives on — upstairs for positive, downstairs for negative. It says nothing at all about whether the number is hot or cold, positive or negative.

$2^{-3}$ is $\dfrac{1}{8}$ , which is small and cheerfully above zero. It is never $-8$ .

The error worth its own slide.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.5 — Definition

Negative Exponent

Definition 1.5.5 — Negative Exponent

If $n$ is an integer and $a \neq 0$ , then

a^{-n} = \dfrac{1}{a^n} \qquad\text{and}\qquad \dfrac{1}{a^{-n}} = a^n

A factor with a negative exponent moves across the fraction bar, and its exponent changes sign on the way.

Only the factor the exponent touches moves: $5y^{-1} = \dfrac{5}{y}$ , while $(5y)^{-1} = \dfrac{1}{5y}$ .

Definition 1.5.5: a negative exponent moves a factor across the fraction bar; the sign of the exponent changes on the way.

7^-3 = -343 is wrong twice over; w^-6 = -6/w^6 is wrong in a third way.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.5 — Definition

Quotient to a Negative Power

Definition 1.5.6 — Quotient to a Negative Power Property

If $a \neq 0$ , $b \neq 0$ and $n$ is an integer, then

\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}

Flip the fraction, and drop the minus sign.

$\left(\dfrac{5}{7}\right)^{-2} = \left(\dfrac{7}{5}\right)^{2} = \dfrac{49}{25}$ . Flipping looks like a trick until you have walked the four-step derivation once.

Definition 1.5.6: a fraction raised to a negative power: flip the fraction, drop the minus sign.

Every shortcut here is a derivation you already did.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.5 — What negative exponents bought

Two things, and both are permanent

The quotient property collapses to one form

No more checking which exponent is larger. Just subtract, and clean up at the end.

\dfrac{a^m}{a^n} = a^{m-n}, \quad a \neq 0

A finished answer carries no negative exponent

If simplifying lands you on $x^{-3}$ , take the extra step and write $\dfrac{1}{x^3}$ . That rewrite is expected on every answer in this section.

$\dfrac{r^5}{r^{-4}} = r^{5-(-4)} = r^{9}$ and $\dfrac{h^3}{h^8} = h^{3-8} = h^{-5} = \dfrac{1}{h^5}$ — one rule, no case split.

The payoff the two-form quotient property was waiting for.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.5 — Worked example

Where the parentheses are decides everything

Example 1.5.3 — Negative exponents on numbers

Simplify $\;(-3)^{-2}$ , $\;-3^{-2}$ , and $\;\left(\dfrac{2}{5}\right)^{-3}$ .

(-3)^{-2} = \dfrac{1}{(-3)^2} = \dfrac{1}{9} \qquad -3^{-2} = -\dfrac{1}{3^2} = -\dfrac{1}{9} \qquad \left(\dfrac{2}{5}\right)^{-3} = \left(\dfrac{5}{2}\right)^{3} = \dfrac{125}{8}

In the first, two negative factors multiply to a positive. In the second the base is $3$ and the minus sign waits outside.

Answers: 1/9; -1/9; 125/8.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.5 — Try It Now

Try It Now 1.5.5

Try It Now 1.5.5 — every answer with only positive exponents

Simplify $\;6^{-2}$ , $\;\dfrac{1}{p^{-5}}$ , $\;\left(\dfrac{4}{9}\right)^{-2}$ , and $\;\dfrac{w^2}{w^7}$ .

6^{-2} = \dfrac{1}{36} \qquad \dfrac{1}{p^{-5}} = p^5 \qquad \left(\dfrac{4}{9}\right)^{-2} = \dfrac{81}{16} \qquad \dfrac{w^2}{w^7} = w^{-5} = \dfrac{1}{w^5}

Part 2 moves upstairs; part 4 subtracts without case-splitting.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.6 — The power properties

Identical groups multiply; mixed piles add

What is $(x^2)^3$ ? The outer exponent says "use the quantity $x^2$ as a factor three times." So write it out.

(x^2)^3 = \underbrace{x^2 \cdot x^2 \cdot x^2}_{3 \text{ factors of } x^2} = \underbrace{x \cdot x \cdot x \cdot x \cdot x \cdot x}_{6 \text{ factors}} = x^6

Three groups of two factors each is $3 \times 2 = 6$ . This is not pooling piles of different sizes — it is copies of one pile, so the counts multiply.

x^2 · x^3 adds to x^5; (x^2)^3 multiplies to x^6. If you cannot recall which, expand.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.6 — Definition

Power Property for Exponents

Definition 1.5.7 — Power Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

(a^m)^n = a^{m \cdot n}

To raise a power to a power, multiply the exponents: $(y^5)^9 = y^{45}$ and $(n^3)^7 = n^{21}$ .

Check with numbers: $(3^2)^3 = 9^3 = 729$ , and $3^{2 \cdot 3} = 3^6 = 729$ .

Definition 1.5.7: a power raised to a power: identical groups of factors, so the counts multiply.

Ten seconds of writing out x's settles this permanently.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.6 — Definition

Product to a Power

Definition 1.5.8 — Product to a Power Property

If $a$ and $b$ are real numbers and $m$ is an integer, then

(ab)^m = a^m b^m

Raise each factor to that power: $(-9d)^2 = 81d^2$ and $(3mn)^3 = 27m^3n^3$ .

A quotient behaves the same way — $\left(\dfrac{a}{b}\right)^m = \dfrac{a^m}{b^m}$ , for $b \neq 0$ . But a sum does not: $(a+b)^2 \neq a^2 + b^2$ , since $7^2 = 49$ and $9 + 16 = 25$ .

Definition 1.5.8: raising a product to a power: the exponent reaches every factor inside, coefficient included.

Chapter 7 takes up what (a+b)^2 actually equals.

Working with Exponents · bookSHelf Integrated Math 1§1.5

Context Pause — the coefficient is a factor too

The 2 in $(2x)^3$ is where most of the errors live

(2x)^3 = 2x \cdot 2x \cdot 2x = 2^3 \cdot x^3 = 8x^3

Writing $2x^3$ leaves the 2 un-raised, and $2x^3 = 2 \cdot x \cdot x \cdot x$ is a genuinely different quantity from $8x^3$ .

Test it at $x = 1$ : the first gives 2, the second gives 8.

Every factor inside got used three times, so every factor picks up the exponent.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.6 — Worked example

Every factor inside the parentheses

Example 1.5.4 — Every factor gets the exponent

Simplify $\;(-2tv)^7\;$ and $\;(4z)^{-3}$ , writing the second answer with only positive exponents.

(-2tv)^7 = (-2)^7 t^7 v^7 = -128t^7v^7

Seven negative factors leave one unpaired negative. In the second, both factors pick up the $-3$ , then cross the bar:

(4z)^{-3} = 4^{-3}z^{-3} = \dfrac{1}{4^3z^3} = \dfrac{1}{64z^3}

Answers: -128t^7v^7 and 1/(64z^3).

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.6 — Try It Now

Try It Now 1.5.6

Try It Now 1.5.6 — power, product, quotient

Simplify $\;(m^4)^6$ , $\;(3ab)^3$ , and $\;\left(\dfrac{2}{7}\right)^2$ .

(m^4)^6 = m^{24} \qquad (3ab)^3 = 27a^3b^3 \qquad \left(\dfrac{2}{7}\right)^2 = \dfrac{4}{49}

The coefficient 3 is a factor inside, so it gets the exponent too.

Answers: m^24; 27a^3b^3; 4/49.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.7 — Putting the properties together

Everything derived so far, collected

Property	Statement	In words
Product	$a^m \cdot a^n = a^{m+n}$	same base multiplied: add exponents
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$	same base divided: subtract exponents
Power	$(a^m)^n = a^{m \cdot n}$	power to a power: multiply exponents
Product to a power	$(ab)^m = a^m b^m$	every factor gets the exponent
Quotient to a power	$\left(\dfrac{a}{b}\right)^m = \dfrac{a^m}{b^m}$	top and bottom each get it
Zero exponent	$a^0 = 1$	anything nonzero to the zero is 1
Negative exponent	$a^{-n} = \dfrac{1}{a^n}$	cross the fraction bar, flip the sign
Quotient to a negative power	$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}$	flip the fraction, drop the minus

Table 1.5.4: denominators nonzero wherever one appears.

Reliable habit: clear the parentheses first, then combine like bases, then clean up negatives and zeros last.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.7 — Worked example

Parentheses first, then like bases

Example 1.5.5 — Two grouped factors at once

Simplify $\;(3x^2y)^4(2xy^2)^3$ .

(3^4x^8y^4)(2^3x^3y^6) = (81x^8y^4)(8x^3y^6)

Regroup by kind, multiply the coefficients, and add the exponents base by base:

81 \cdot 8 \cdot x^{8+3} \cdot y^{4+6} = 648x^{11}y^{10}

Answer: 648x^11 y^10.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.7 — Worked example

Nearly everything at once

Example 1.5.6 — Every property in one problem

Simplify $\;\left(\dfrac{3z^{-2}}{w^{2}}\right)^{-2}$ , writing the answer with only positive exponents.

\dfrac{(3z^{-2})^{-2}}{(w^2)^{-2}} = \dfrac{3^{-2}z^{4}}{w^{-4}}

In the numerator $(-2)(-2) = 4$ ; in the denominator $(2)(-2) = -4$ . Now move the negatives across the bar — $3^{-2}$ goes down, $w^{-4}$ comes up:

\dfrac{z^4 w^4}{3^2} = \dfrac{z^4w^4}{9}

Answer: z^4 w^4 / 9.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.7 — The errors this section is designed to prevent

Every one of these is a rule applied to the wrong thing

Expression	Correct	Common error	Diagnosis
$(x^4)^3$	$x^{12}$	$x^7$	added exponents where the rule multiplies
$(2x)^4$	$16x^4$	$2x^4$	exponent not applied to the 2
$5x^{-2}$	$\dfrac{5}{x^2}$	$\dfrac{1}{5x^2}$	moved the 5, which had no negative exponent
$x^{-3}$	$\dfrac{1}{x^3}$	$-x^3$	treated the negative exponent as a sign
$(-4)^0$	$1$	$0$	zero exponent gives 1, not 0
$-3^2$	$-9$	$9$	base is 3; the minus is not inside

Table 1.5.5: when a rule fails, ask which thing it was applied to.

Expanding settles every row on this table.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.7 — Try It Now

Try It Now 1.5.7

Try It Now 1.5.7 — every answer with only positive exponents

Simplify $\;(a^2)^5(a^3)^4$ , then $\;(-2p^3q^4)^3$ , then $\;(4t^{-2})^2$ .

(a^2)^5(a^3)^4 = a^{10} \cdot a^{12} = a^{22} \qquad (-2p^3q^4)^3 = -8p^9q^{12}

(4t^{-2})^2 = 16t^{-4} = \dfrac{16}{t^4}

Only the $t$ has a negative exponent, so only the $t$ moves down.

Answers: a^22; -8p^9q^12; 16/t^4.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.8 — Scientific notation

What the algebra is for when the quantities get out of hand

A number in the millions carries a long tail of zeros; a number in the millionths carries a long head of them. Because our number system is built on tens, powers of ten give a compact way out.

Number	As a product	Power of ten	Result
$4{,}000$	$4 \times 1{,}000$	$1{,}000 = 10^3$	$4 \times 10^3$
$0.004$	$4 \times \dfrac{1}{1{,}000}$	$\dfrac{1}{10^3} = 10^{-3}$	$4 \times 10^{-3}$

Table 1.5.6: the second row is where the negative-exponent work pays off directly.

Without §1.5.5, $10^{-3}$ would be a symbol with no meaning; with it, small numbers get written as compactly as large ones.

Powers of ten are just powers — nothing about a base of 10 is special.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.8 — Definition

Scientific Notation

Definition 1.5.9 — Scientific Notation

A number is in scientific notation when it is written as

a \times 10^{n}, \qquad 1 \leq |a| < 10, \ n \text{ an integer.}

Digits in front, scale in the exponent — so you can read a quantity's size at a glance and never miscount a row of zeros.

The requirement is strict: $37 \times 10^3$ is a true statement about a number but is not scientific notation, because 37 is not between 1 and 10.

Definition 1.5.9: scientific notation splits a quantity into its digits and its scale.

The multiplication sign is written as a cross here, even though algebra otherwise avoids it.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.8 — Worked example

Decimal form to scientific notation

Example 1.5.7 — A number greater than 1

Write $37{,}000$ in scientific notation. Move the point so one nonzero digit sits to its left, count the places, and attach the power of ten.

1 — $37{,}000$ is greater than 1, so the power of ten will be positive. 2 — move the point to $3.7$ . 3 — the move was 4 places.

37{,}000 = 3.7 \times 10^{4}

Check: $3.7 \times 10{,}000 = 37{,}000$ . More of the same: $0.022 = 2.2 \times 10^{-2}$ and $0.00000654 = 6.54 \times 10^{-6}$ .

A number between 0 and 1 is small, so its power of ten is negative.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.8 — Worked example

Scientific notation back to decimal form

Example 1.5.8 — Both directions of the decimal point

Write $6.2 \times 10^{3}$ and $-8.9 \times 10^{-2}$ in decimal form. Read the exponent, then move the point that many places — right if positive, left if negative.

6.2 \times 10^{3} = 6{,}200 \qquad\qquad -8.9 \times 10^{-2} = -0.089

Two zeros had to be supplied as placeholders. And keep the two minus signs straight: the one in front says the number is negative; the one in the exponent says it is small.

Answers: 6,200 and -0.089.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.8 — What scientific notation is for

The exponent does most of the comparing

Sorting by size

Compare exponents first; only when two numbers share an exponent do you look at the leading factors. Every one of those comparisons is between numbers under 10 — easier than comparing the full quantities.

Multiplying and dividing

Group the leading factors together and the powers of ten together, then use the product or quotient property on the tens.

(3 \times 10^{5})(2 \times 10^{-8}) = 6 \times 10^{-3} \qquad\qquad \dfrac{9 \times 10^{-3}}{3 \times 10^{2}} = 3 \times 10^{-5}

If the leading factors multiply to 10 or more, rewrite that factor too: $(4 \times 10^{7})(5 \times 10^{6}) = 20 \times 10^{13} = 2 \times 10^{14}$ .

Exactly the separation §1.5.2 insisted on: coefficients multiply, exponents add.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.8 — Try It Now

Try It Now 1.5.8

Try It Now 1.5.8 — both directions, then a product

Write $0.00046$ in scientific notation, write $3.05 \times 10^{4}$ in decimal form, and multiply $\;(5 \times 10^{-6})(3 \times 10^{9})$ .

0.00046 = 4.6 \times 10^{-4} \qquad 3.05 \times 10^{4} = 30{,}500

The leading factor 15 is not between 1 and 10, so rewrite it and combine again:

(5 \times 10^{-6})(3 \times 10^{9}) = 15 \times 10^{3} = 1.5 \times 10^{4}

Answers: 4.6 × 10^-4; 30,500; 1.5 × 10^4.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.9 — Why this section comes before Chapter 4

Where the zero and negative exponents stop being abstract

Chapter 4 studies exponential functions — a quantity repeatedly multiplied by the same factor, written $y = a \cdot b^{t}$ , with the variable sitting in the exponent.

$t = 0$ returns the starting value

A coral structure measured at 1,200 cm³ doubles each year: $y = 1200 \cdot 2^{t}$ . At $t = 0$ , $1200 \cdot 2^0 = 1200$ — the zero exponent property is what makes that work.

Negative $t$ reads backwards in time

$y = 1200 \cdot 2^{-1} = 1200 \cdot \dfrac{1}{2} = 600$ — the coral a year earlier. The negative exponent is not a strange edge case; it is how the model talks about the past.

Chapter 4's question will be about growth, not about whether exponents add or multiply.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5.9 — Try It Now

Try It Now 1.5.9

Try It Now 1.5.9 — and name the property each one used

A colony is modeled by $p = 6400 \cdot 2^{t}$ , where $t$ is years since the first count. Find the population at $t = 0$ , at $t = -3$ , and at $t = 2$ .

p(0) = 6400 \cdot 2^{0} = 6400 \qquad p(-3) = 6400 \cdot \dfrac{1}{2^3} = 800 \qquad p(2) = 6400 \cdot 4 = 25{,}600

Zero exponent property; definition of a negative exponent; the plain definition of an exponent.

Three years before the first count the colony was 800 — as a doubling model run backwards should give.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5 — Key Terms

Key Terms

base — in $a^m$ , the number $a$ used repeatedly as a factor.

exponent — in $a^m$ , the number $m$ counting how many times the base is used.

exponential notation — the shorthand $a^m$ for $m$ identical factors of $a$ .

product property — same base multiplied: keep the base, add the exponents.

quotient property — same base divided: keep the base, subtract the exponents.

zero exponent property — any nonzero base to the zero power equals 1.

negative exponent — a sign that sends its factor across the fraction bar: $a^{-n} = \dfrac{1}{a^n}$ .

power property — to raise a power to a power, multiply the exponents.

product to a power property — raise every factor inside to that power.

quotient to a power property — raise numerator and denominator to that power.

quotient to a negative power property — flip the fraction, take the matching positive power.

scientific notation — $a \times 10^{n}$ with $1 \leq |a| < 10$ and $n$ an integer.

All twelve section terms.

Working with Exponents · bookSHelf Integrated Math 1§1.5

§1.5 — The headline result

Every property in this section is a fact about counting factors

Not one of them was announced. Expand, count, collapse — and the rule that comes out is the only rule that could have.

$a^m \cdot a^n$ pools two piles, so the counts add. $(a^m)^n$ copies one pile, so the counts multiply. $a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$ are what the divide-by-the-base staircase forces when you refuse to break its pattern.

† Which is why forgetting one is recoverable: write the factors out and count them. Ten seconds settles it permanently.

The section's actual thesis — the property table is the residue, not the argument.

Working with Exponents · bookSHelf Integrated Math 1§1.5

1.5

§1.5 — Conclusions

What to carry forward

The one idea

An exponent is a tally of factors. Pooling piles adds the counts, copying a pile multiplies them, pairing off subtracts them, and continuing the pattern past $a^1$ forces $a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$ . Scientific notation is that whole apparatus pointed at numbers too big or too small to write.

Where it goes wrong

Swapping the two jobs in $3x^4 \cdot 5x^2$ — coefficients multiply, exponents add. Reading $2^{-3}$ as $-8$ instead of $\dfrac{1}{8}$ . Leaving the coefficient un-raised in $(2x)^3$ . Losing track of what the parentheses make the base.

Next: Chapter 2 — and eventually Chapter 4, where $t$ moves into the exponent for good. Back to start.

Closes Chapter 1: the chapter began with patterns and closes with the algebra those functions are made of.

Working with Exponents

Objectives

The exponent is a count, and parentheses decide what is counted

The raised number is not something you do

Exponential Notation

Try It Now 1.5.1

Nothing was decided; the count decided it

Product Property for Exponents

Coefficients multiply; exponents add

Naming the reason behind each step

Try It Now 1.5.2

Every factor downstairs takes one upstairs off the floor

Quotient Property for Exponents

Split the number part from the variable part

Try It Now 1.5.3

The value of 303^030 is not invented; it is forced

Zero Exponent Property

You do not get to invent a step height

Try It Now 1.5.4

The staircase has no reason to stop at 303^030

It says which floor, never which side of zero

Negative Exponent

Quotient to a Negative Power

Two things, and both are permanent

The quotient property collapses to one form

A finished answer carries no negative exponent

Where the parentheses are decides everything

Try It Now 1.5.5

Identical groups multiply; mixed piles add

Power Property for Exponents

Product to a Power

The 2 in (2x)3(2x)^3(2x)3 is where most of the errors live

Every factor inside the parentheses

Try It Now 1.5.6

Everything derived so far, collected

Parentheses first, then like bases

Nearly everything at once

Every one of these is a rule applied to the wrong thing

Try It Now 1.5.7

What the algebra is for when the quantities get out of hand

Scientific Notation

Decimal form to scientific notation

Scientific notation back to decimal form

The exponent does most of the comparing

Sorting by size

Multiplying and dividing

Try It Now 1.5.8

Where the zero and negative exponents stop being abstract

t=0t = 0t=0 returns the starting value

Negative ttt reads backwards in time

Try It Now 1.5.9

Key Terms

Every property in this section is a fact about counting factors

What to carry forward

The one idea

Where it goes wrong

The value of $3^0$ is not invented; it is forced

The staircase has no reason to stop at $3^0$

The 2 in $(2x)^3$ is where most of the errors live

$t = 0$ returns the starting value

Negative $t$ reads backwards in time