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Virginia SOL Mathematics Textbook

Algebra 1 Workbook — Chapter 10: The Laws of Exponents

SOL A.EO.3 (a, b) · Companion to Textbook Chapter 10

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 126.

PAGE 1 — Chapter opener

Chapter 10 · The Laws of Exponents

Standard A.EO.3 (a, b)

In this chapter you will:

Words to know: base · exponent · power · product law · quotient law · power of a power · power of a product · power of a quotient · zero exponent · negative exponent · reciprocal · coefficient · monomial · ratio of monomials · integer exponent · order of magnitude · scientific notation

Convention: the exponent reaches only the symbol it sits on. −32=−9-3^2 = -9, but (−3)2=9(-3)^2 = 9. In 3x23x^2 only the xx is squared; in (3x)2(3x)^2 both factors are.

Convention: a negative exponent is not a negative number. 2−3=182^{-3} = \tfrac18, a positive number smaller than 11. Final answers use positive exponents.

Convention: every law derived from a quotient carries a≠0a \neq 0, and 000^0 is undefined.

Calculator. Algebra 1 has no no-calculator standards. Use one to confirm: entering 2−32^{-3} and seeing 0.1250.125 is one keystroke and it catches a sign error.

PAGE 2 — Counting factors

10.1 Powers and the Product Law

FIGURE: fig1-product-rule-expansion.png (full width)

Fill in the blanks.

In ana^n, the number aa is the ____________ and the number nn is the ____________.

a1=a^1 = ______, because ____________ copy of aa is aa.

In the figure, the blue brace counts ______ factors, the red brace counts ______ factors, and the black brace counts ______ factors in all.

The product law: am⋅an=a‾a^m \cdot a^n = a^{\underline{\hspace{2cm}}}

  1. a3⋅a2a^3 \cdot a^2 — row of factors: _______________________ Braces count: ______, ______, ______ Single power: ______

  2. Shaded row, a2⋅a6a^2 \cdot a^6 — factors written out: _______________________

    Count: ______________ Single power: ______

  3. Explain. Why do the exponents add? Use counting, and do not use the word "rule."

PAGE 3 — Practice · the product law

Practice · Adding the Counts

  1. 23⋅24=2^3 \cdot 2^4 = ______ Check: original == ______ answer == ______

  2. m5⋅m7=m^5 \cdot m^7 = ______

  3. Write each as a single power.

Product Single power
a) x6⋅x3x^6 \cdot x^3
b) 72⋅757^2 \cdot 7^5
c) b⋅b8b \cdot b^8
d) y4⋅y4y^4 \cdot y^4
  1. Simplify.
Product Coefficients Exponents Answer
a) 3x2⋅5x63x^2 \cdot 5x^6
b) −4a3⋅6a5-4a^3 \cdot 6a^5
c) 2p7⋅p2⋅3p2p^7 \cdot p^2 \cdot 3p
d) (−c5)(−8c4)(-c^5)(-8c^4)

PAGE 4 — Practice · two and three variables

Practice · One Base at a Time

  1. Simplify each multivariable product.
Product Answer
a) x3y2⋅x4y5x^3y^2 \cdot x^4y^5
b) 2m2n⋅5m6n32m^2n \cdot 5m^6n^3
c) −3a4b2⋅a2b7-3a^4b^2 \cdot a^2b^7
d) 4r2s3t⋅3rs2t44r^2s^3t \cdot 3rs^2t^4

  1. Explain. Why can x3⋅y4x^3 \cdot y^4 not be written as a single power? What would (xy)7(xy)^7 mean?

  2. Explain. Why is a1=aa^1 = a? Why would the product law fail on b⋅b8b \cdot b^8 if you refused to read bb as b1b^1?

  3. Find each missing exponent.

    a) x4⋅x‾=x11x^4 \cdot x^{\underline{\hspace{2cm}}} = x^{11}

    b) 3a‾⋅4a5=12a93a^{\underline{\hspace{2cm}}} \cdot 4a^5 = 12a^9

    c) m2n‾⋅m5n3=m7n8m^2n^{\underline{\hspace{2cm}}} \cdot m^5n^3 = m^7n^8

PAGE 5 — Where the minus sign lives

Which Symbol Has the Exponent?

FIGURE: fig7-negative-base-versus-negated-power.png (full width)

Complete the frame. The ____________________ decide what the base is. Read −an-a^n as "the ____________ of ana^n."

  1. (−3)2=(-3)^2 = ______ −32=-3^2 = ______ What makes them different?

  2. Fill in all four, then answer the question.

Expression Value
(−2)5(-2)^5
−25-2^5
(−2)6(-2)^6
−26-2^6 
When do the parentheses matter? _______________________________________

  1. Find the error. A student writes −52=25-5^2 = 25.

    What went wrong? _______________________________________

    Correct value: ______ Expression whose value is 2525: ____________

PAGE 6 — Application and exit ticket 10.1

Apply and Check

  1. Apply it. An archive holds 252^5 files, each exactly 2122^{12} bytes.

    Total bytes as one power of 22: ______ As an ordinary number: ______________

    Law used: _______________________

  2. Find the error. A student writes x3⋅x4=x12x^3 \cdot x^4 = x^{12}.

    What went wrong? _______________________ Correct answer: ______

    Count of factors that settles it: _______________________

Exit Ticket · Lesson 10.1

Name: ________________________ Date: ____________

  1. y7⋅y6=y^7 \cdot y^6 = ______

  2. −2x3y⋅7x5y4=-2x^3y \cdot 7x^5y^4 = _______________________

  3. (−4)2=(-4)^2 = ______ −42=-4^2 = ______ Which symbol carries the exponent in each? ____________________

  4. Explain. Why must the two bases match? Use x3⋅y4x^3 \cdot y^4.

PAGE 7 — Matching factors off

10.2 The Quotient Law

FIGURE: fig2-quotient-rule-cancellation.png (full width)

Complete the frame. Each pair struck through is a factor divided by ____________, which equals ______.

The quotient law: aman=a‾\dfrac{a^m}{a^n} = a^{\underline{\hspace{2cm}}}, for a‾0a \underline{\hspace{1cm}} 0

The subtraction is always ____________________ exponent minus ____________________ exponent.

  1. x5x2\dfrac{x^5}{x^2} — all factors shown: _______________________

    Pairs struck through: ______ Factors left over: ______ Single power: ______

  2. Shaded row, y7y4\dfrac{y^7}{y^4} — pairs canceled: ______ yys remaining: ______ Single power: ______

  3. 2522=\dfrac{2^5}{2^2} = ______ 23=2^3 = ______ What the agreement confirms: _______________________

PAGE 8 — Practice · the quotient law

Practice · Subtracting the Counts

  1. m9m4=\dfrac{m^9}{m^4} = ______

  2. 12a73a2=\dfrac{12a^7}{3a^2} = ______

  3. Explain. Why must the law say a≠0a \neq 0? What goes wrong at a=0a = 0?

  4. Write each as a single power.

Quotient Single power
a) x10x4\dfrac{x^{10}}{x^4}
b) 5853\dfrac{5^8}{5^3}
c) n6n\dfrac{n^6}{n}
d) c12c5\dfrac{c^{12}}{c^5}
  1. Simplify.
Quotient Coefficients Exponents Answer
a) 20x84x3\dfrac{20x^8}{4x^3}
b) −18a66a2\dfrac{-18a^6}{6a^2}
c) 15m925m4\dfrac{15m^9}{25m^4}
d) −24p7−8p3\dfrac{-24p^7}{-8p^3}

PAGE 9 — Practice · ratios of monomials

Practice · One Subtraction per Base

  1. Simplify each ratio of monomials.
Ratio Base by base Answer
a) x7y5x3y2\dfrac{x^7y^5}{x^3y^2}
b) 12a6b44a2b\dfrac{12a^6b^4}{4a^2b}
c) −30m8n65m3n2\dfrac{-30m^8n^6}{5m^3n^2}
d) r9s4t2r4s2t\dfrac{r^9s^4t^2}{r^4s^2t}

  1. Find each missing exponent.

    a) x‾x3=x9\dfrac{x^{\underline{\hspace{2cm}}}}{x^3} = x^9

    b) 20m75m‾=4m3\dfrac{20m^7}{5m^{\underline{\hspace{2cm}}}} = 4m^3

    c) a8b‾a3b4=a5b2\dfrac{a^8b^{\underline{\hspace{2cm}}}}{a^3b^4} = a^5b^2

PAGE 10 — Reasoning and errors

Why It Works, and Where It Breaks

  1. Explain. Why does subtracting exponents do the same job as canceling matched pairs? Use x5x2\dfrac{x^5}{x^2}, and say what each canceled pair is worth.

  2. Find the error. A student writes x8x2=x4\dfrac{x^8}{x^2} = x^4.

    What did the student do to the exponents? _______________________ Correct answer: ______

    Leftover-factor check: _______________________

  3. Find the error. A student writes 12a64a2=8a4\dfrac{12a^6}{4a^2} = 8a^4.

    What went wrong? _______________________ Correct answer: ______

    Coefficients get ____________; exponents get ____________.

  4. Explain. In aman\dfrac{a^m}{a^n} with m>nm > n, the leftover count is positive. Using leftover factors only, describe what happens when m=nm = n, and when m<nm < n.

    m=nm = n: _______________________________________

    m<nm < n: _______________________________________

PAGE 11 — Application and exit ticket 10.2

Apply It · Comparing Sizes

  1. Apply it. A stadium seats about 3×1043 \times 10^4; an auditorium seats about 3×1023 \times 10^2.

    Ratio as a power of ten: ______ In words: _______________________________________

  2. Apply it. A drive holds 2402^{40} bytes; each backup image is 2282^{28} bytes.

    Images that fit, as a power of 22: ______ As an ordinary number: ______________

Exit Ticket · Lesson 10.2

Name: ________________________ Date: ____________

  1. x11x6=\dfrac{x^{11}}{x^6} = ______

  2. −28a9b57a4b2=\dfrac{-28a^9b^5}{7a^4b^2} = _______________________

  3. 21227=\dfrac{2^{12}}{2^7} = ______ as a power of 22, and ______ as a number

  4. Explain. Why does every statement of the quotient law carry a≠0a \neq 0?

PAGE 12 — The pattern down to zero

10.3 Zero and Negative Exponents

FIGURE: fig3-quotient-pattern-forces-zero-exponent.png (full width)

Complete the frame. Step down one exponent, ____________ by 22. The last row is not a ____________.

  1. Complete the table from the figure.
Power Value From the row above
242^4
232^3
222^2
212^1
202^0

  1. Two readings of a3a3\dfrac{a^3}{a^3}.

    By the quotient law: =a‾=a‾= a^{\underline{\hspace{2cm}}} = a^{\underline{\hspace{1cm}}}

    By canceling all three pairs: == ______

    Therefore a0=a^0 = ______, for a‾0a \underline{\hspace{1cm}} 0. Why can it not be otherwise?

  2. 70=7^0 = ______ (−5)0=(-5)^0 = ______ (3x)0=(3x)^0 = ______ (for x≠0x \neq 0)

PAGE 13 — A zero exponent you can count

The Hour Nothing Has Happened Yet

FIGURE: fig9-cell-division-doubling-table.png (full width)

  1. Which column is boxed? ____________ Cells recorded there: ______ Power of 22 there: ______

    Why is that column a real moment, not a technicality? _______________________________________

  2. Apply it. Use the same figure.

    Why does the hour-00 column record 11 cell? _______________________________________

    Cells after 66 hours: ______ After 1010 hours: ______ as a power of 22, ______________ as a number

  3. Explain. Why do both a0=1a^0 = 1 and a−n=1ana^{-n} = \tfrac{1}{a^n} require a≠0a \neq 0? Why is 000^0 undefined?

  4. Explain. Name the two patterns that disagree at 000^0, and say why the derivation of a0=1a^0 = 1 cannot settle a=0a = 0.

PAGE 14 — The pattern continued below zero

Nothing New Is Decided at 202^0

FIGURE: fig4-descending-pattern-forces-negative-exponents.png (full width)

Complete the frame. A negative exponent is not a ____________________. It is an instruction to take a ____________________.

  1. Complete the three shaded rows.
Power Value As a decimal From the row above
2−12^{-1}
2−22^{-2}
2−32^{-3}

  1. Two readings of a2a5\dfrac{a^2}{a^5}.

    By the quotient law: =a‾=a‾= a^{\underline{\hspace{2cm}}} = a^{\underline{\hspace{1cm}}}

    By canceling the two pairs: == ______

    Therefore a−n=a^{-n} = ______, for a‾0a \underline{\hspace{1cm}} 0

  2. Explain. Using the divide-by-22 column, why must 2−12^{-1} be 12\tfrac12 and not −2-2? What would have to happen to the pattern for −2-2 to be right?

PAGE 15 — Practice · evaluating

Practice · Exact Values

  1. Evaluate exactly.
Power Value
a) 909^0
b) 4−24^{-2}
c) (−3)−3(-3)^{-3}
d) (25)−1\left(\tfrac{2}{5}\right)^{-1}
  1. Give each as a decimal, then confirm it on a calculator.
Power Decimal What you entered
a) 2−32^{-3}
b) 5−25^{-2}
c) 10−410^{-4}

  1. Rewrite with positive exponents only.

    a) x−5=x^{-5} = ____________ b) 3y−2=3y^{-2} = ____________

    c) 1m−4=\dfrac{1}{m^{-4}} = ____________ d) a−3b−2=\dfrac{a^{-3}}{b^{-2}} = ____________

PAGE 16 — Practice · combining laws

Practice · Signs in the Exponents

  1. Simplify, positive exponents.
Expression Answer
a) x−3⋅x7x^{-3} \cdot x^7
b) m2m6\dfrac{m^2}{m^6}
c) a−4⋅a−3a^{-4} \cdot a^{-3}
d) y−2y5\dfrac{y^{-2}}{y^5}
  1. Simplify each multivariable expression, positive exponents.
Expression Base by base Answer
a) x−2y5⋅x6y−3x^{-2}y^5 \cdot x^6y^{-3}
b) a3b−2a−1b4\dfrac{a^3b^{-2}}{a^{-1}b^4}
c) 6m−3n42m2n−1\dfrac{6m^{-3}n^4}{2m^2n^{-1}}
d) 4p0q−2⋅5p3q24p^0q^{-2} \cdot 5p^3q^2

  1. Explain. Why is 3x−2=3x23x^{-2} = \dfrac{3}{x^2} and not 13x2\dfrac{1}{3x^2}?

    At x=2x = 2: correct form gives ______ wrong form gives ______

PAGE 17 — Reasoning, errors, application

Forced, Not Chosen

  1. Explain. Why is a0=1a^0 = 1 forced rather than chosen? Use the two readings of a3a3\dfrac{a^3}{a^3}, and say where a≠0a \neq 0 enters.

  2. Find the error. A student says 5−2=−255^{-2} = -25.

    Two separate mistakes: _______________________________________

    Correct value: ______ Expression that really equals −25-25: ____________

  3. Find the error. A student says x0=0x^0 = 0 "because the exponent is zero."

    What went wrong? _______________________ Correct value: ______

    Quotient that forces it: ____________

  4. Apply it. A thumbnail image is 2−32^{-3} megabytes.

    As a fraction: ______ As a decimal: ______ Thumbnails per megabyte: ______

PAGE 18 — Exit ticket 10.3

Exit Ticket · Lesson 10.3

Name: ________________________ Date: ____________

  1. 120=12^0 = ______ 3−4=3^{-4} = ______ (−2)−4=(-2)^{-4} = ______

  2. 5x−3y−2=\dfrac{5x^{-3}}{y^{-2}} = _______________________

  3. a2b−5a−3b−1=\dfrac{a^2b^{-5}}{a^{-3}b^{-1}} = _______________________

  4. 23=82^3 = 8, 22=42^2 = 4, 21=22^1 = 2, 20=‾2^0 = \underline{\hspace{2cm}}, 2−1=‾2^{-1} = \underline{\hspace{2cm}}

    The single rule producing each entry from the one above: _______________________________________

  5. Explain. In two sentences, why is a0=1a^0 = 1 forced by the quotient law?

PAGE 19 — Counting groups

10.4 Powers of Bases

FIGURE: fig5-power-of-a-power-expansion.png (full width)

Complete the frame. The outer exponent counts ____________. The inner exponent counts ____________ inside a group. So the total ____________________.

Power of a power: (am)n=a‾(a^m)^n = a^{\underline{\hspace{2cm}}}

  1. (a2)3(a^2)^3 — three groups: _______________________ Brace counts: _______________________ Single power: ______

  2. Shaded row, (x4)3(x^4)^3 — groups: ______ factors in each: ______ Single power: ______

  3. (23)2=(2^3)^2 = ______2=^2 = ______ and 26=2^6 = ______ What the agreement confirms: _______________________

  4. Explain. Why does (a2)3(a^2)^3 multiply the exponents while a2⋅a3a^2 \cdot a^3 adds them?

    (a2)3=(a^2)^3 = ______ a2⋅a3=a^2 \cdot a^3 = ______

    The two pictures: _______________________________________

PAGE 20 — Products and quotients inside parentheses

Everything Inside Takes the Exponent

FIGURE: fig6-power-of-a-product-and-quotient.png (full width)

  1. Left panel — write the three lines of the derivation of (2x)3(2x)^3.

    The boxed law: (ab)n=(ab)^n = ____________

  2. Right panel — write the derivation of (xy)4\left(\dfrac{x}{y}\right)^4.

    The boxed law: (ab)n=\left(\dfrac{a}{b}\right)^n = ____________, for b‾0b \underline{\hspace{1cm}} 0

    Why the restriction? _______________________________________

Warning frame. (3x)4=(3x)^4 = ____________, not 3x43x^4. The ____________________ takes the exponent too.

PAGE 21 — Practice · powers of bases

Practice · Distribute the Outer Exponent

  1. Simplify.
Expression Answer
a) (m5)4(m^5)^4
b) (24)3(2^4)^3
c) (y7)2(y^7)^2
d) ((a2)3)2\big((a^2)^3\big)^2
  1. Simplify.
Expression Coefficient Variables Answer
a) (3x)4(3x)^4
b) (−2a)5(-2a)^5
c) (5m2n)3(5m^2n)^3
d) (−4p3q2)2(-4p^3q^2)^2

  1. Simplify.

    a) (x3)3=\left(\dfrac{x}{3}\right)^3 = ____________ b) (2ab)4=\left(\dfrac{2a}{b}\right)^4 = ____________

    c) (m3n2)5=\left(\dfrac{m^3}{n^2}\right)^5 = ____________ d) (3x2y4)2=\left(\dfrac{3x^2}{y^4}\right)^2 = ____________

PAGE 22 — Negative outer exponents, errors, application

Practice · A Negative on the Outside

  1. Simplify with positive exponents.

    a) (x4)−3=(x^4)^{-3} = ____________ b) (2m)−3=(2m)^{-3} = ____________

    c) (ab)−2=\left(\dfrac{a}{b}\right)^{-2} = ____________ d) (3x−2)−1=(3x^{-2})^{-1} = ____________

  2. Find the error. A student writes (3x)2=3x2(3x)^2 = 3x^2.

    What went wrong? _______________________ Correct answer: ____________

    At x=1x = 1: correct gives ______ student's gives ______

  3. Find the error. A student writes (x3)4=x7(x^3)^4 = x^7.

    Which law did the student use? _______________________ Correct answer: ______

    Correct count of factors: _______________________

  4. Find each missing exponent.

    a) (x‾)5=x20\left(x^{\underline{\hspace{2cm}}}\right)^5 = x^{20}

    b) (2a3)‾=8a9\left(2a^3\right)^{\underline{\hspace{2cm}}} = 8a^9

    c) (m4n)3=m‾n‾\left(\dfrac{m^4}{n}\right)^3 = \dfrac{m^{\underline{\hspace{2cm}}}}{n^{\underline{\hspace{2cm}}}}

PAGE 23 — Application and exit ticket 10.4

Apply It · Volumes and Areas

  1. Apply it. A storage cube has edge 2x32x^3 centimeters.

    Volume: _______________________ Laws used: _______________________________________

  2. Apply it. One kilometer is 10310^3 meters.

    Square meters in one square kilometer, as a power of ten: ______ As a number: ______________

    Law used: _______________________

  3. Explain. Why is (ab)n=anbn(ab)^n = a^nb^n true but (a+b)n=an+bn(a+b)^n = a^n + b^n false?

    At a=3a = 3, b=4b = 4, n=2n = 2: left side == ______ right side == ______

    Which step of the derivation fails for a sum? _______________________________________

Exit Ticket · Lesson 10.4

  1. (a6)3=(a^6)^3 = ______

  2. (−3m2n5)3=(-3m^2n^5)^3 = _______________________

  3. (2x3y2)4=\left(\dfrac{2x^3}{y^2}\right)^4 = _______________________

  4. Explain. The difference between x3⋅x5x^3 \cdot x^5 and (x3)5(x^3)^5.

    x3⋅x5=x^3 \cdot x^5 = ______ (x3)5=(x^3)^5 = ______ What is counted in each: _______________________

PAGE 24 — Putting the laws together

10.5 Multivariable Expressions and Ratios of Monomials

Complete the definition. A monomial is a ____________, a ____________, or a ____________ of numbers and variables.

An answer is simplified when: each base appears ____________ once · every exponent is ____________ · the coefficient is a single number or a fraction in ____________________ · no power of a power is left uncleared.

The order of the work.

  1. Clear every ____________________ first.

  2. Multiply within the ____________________ and within the ____________________.

  3. Divide the ____________________ and reduce.

  4. Subtract exponents ____________ by ____________.

  5. Move any ____________ exponent across the bar.

  6. 15x5y35x2y7=\dfrac{15x^5y^3}{5x^2y^7} = _______________________

  7. (2a3)48a5=\dfrac{(2a^3)^4}{8a^5} = _______________________ Step that must come first: _______________________

  8. 6m4n−29m−1n3=\dfrac{6m^4n^{-2}}{9m^{-1}n^3} = _______________________

    Subtraction on the base mm, with parentheses: _______________________

  9. What is a monomial? Is 3x2y\dfrac{3x^2}{y} one? ______ Why?

PAGE 25 — Practice · ratios of monomials

Practice · Simplify Completely

  1. Simplify, positive exponents.
Ratio Coefficients Base by base Answer
a) x8y2x3y6\dfrac{x^8y^2}{x^3y^6}
b) 24a5b318a2b7\dfrac{24a^5b^3}{18a^2b^7}
c) −14m3n87m7n2\dfrac{-14m^3n^8}{7m^7n^2}
d) p4q−3rp−2q2r5\dfrac{p^4q^{-3}r}{p^{-2}q^2r^5}
  1. Simplify, positive exponents.
Expression Answer
a) (3x2y)3⋅2xy4(3x^2y)^3 \cdot 2xy^4
b) (2m3)4(4m2)2\dfrac{(2m^3)^4}{(4m^2)^2}
c) (a4ba2b3)3\left(\dfrac{a^4b}{a^2b^3}\right)^3
d) 5x−2y320x4y−1\dfrac{5x^{-2}y^3}{20x^4y^{-1}}

PAGE 26 — Practice · three variables

Practice · Three Bases at Once

  1. Simplify, positive exponents.
Expression Answer
a) 18r6s−2t412r2s3t4\dfrac{18r^6s^{-2}t^4}{12r^2s^3t^4}
b) (2rs2)3t4r2s4t−2\dfrac{(2rs^2)^3 t}{4r^2s^4t^{-2}}
c) −x3y5z2x5y2z−1\dfrac{-x^3y^5z^2}{x^5y^2z^{-1}}
d) (3a2b)2(3ab2)2\dfrac{(3a^2b)^2}{(3ab^2)^2}

  1. Explain. For (2x3)48x5\dfrac{(2x^3)^4}{8x^5}, list the laws you use in order.

    1. _______________________ 2. _______________________ 3. _______________________

    Simplified result: ____________

  2. Explain. Why are x3x7\dfrac{x^3}{x^7}, x−4x^{-4}, and 1x4\dfrac{1}{x^4} three names for one quantity? Which does this chapter report, and why?

PAGE 27 — Orders of magnitude

Powers of Ten

FIGURE: fig8-orders-of-magnitude-scale.png (full width)

Complete the frame. One step right ____________________ by 1010. One step left ____________________ by 1010.

At 10010^0 the value is ______, labeled in the figure as ____________________.

  1. One step right does: _______________________ One step left does: _______________________

    Value at 10010^0: ______ Everyday length there: _______________________

  2. 11 mm and 11 km — ratio as a quotient of powers of ten: ____________ Simplified: ______

    In words: _______________________________________

  3. Apply it. A credit card is about 10−310^{-3} m thick; Richmond to Chicago is about 10610^6 m.

    Ratio: ____________ Simplified: ______ Orders of magnitude apart: ______

PAGE 28 — Scientific notation in context

Apply It · Comparing Very Large and Very Small

Frame. To divide two numbers in scientific notation: divide the ____________________, then ____________ the exponents.

  1. Apply it. United States about 3.4×1083.4 \times 10^8 people; a small town about 1.7×1041.7 \times 10^4.

    Front numbers: 3.41.7=\dfrac{3.4}{1.7} = ______ Exponents: 10‾=10‾10^{\underline{\hspace{2cm}}} = 10^{\underline{\hspace{1cm}}}

    Answer: ____________ In words: _______________________________________

  2. Apply it. A video is 4×1094 \times 10^9 bytes; a photo is 2×1062 \times 10^6 bytes.

    Photos per video, in scientific notation: ____________ As an ordinary number: ______________

PAGE 29 — Errors, checking, exit ticket 10.5

Find the Error, Then Check Your Own

  1. Find the error. A student writes 24a5b318a2b7=4a3b43\dfrac{24a^5b^3}{18a^2b^7} = \dfrac{4a^3b^4}{3}.

    Error on the base bb: _______________________ Correct answer: ____________

    The rule about order: _______________________________________

  2. Find the error. A student writes 6m49m−1=2m33\dfrac{6m^4}{9m^{-1}} = \dfrac{2m^3}{3}.

    What went wrong? _______________________ Correct answer: ____________

    The subtraction: _______________________

  3. Check it. Is x8y2x3y6=x5y4\dfrac{x^8y^2}{x^3y^6} = \dfrac{x^5}{y^4}?

Values Original Simplified
x=2x = 2, y=2y = 2
x=3x = 3, y=1y = 1 
 Why is agreement at two pairs **not** a proof? _______________________________________

 Why would disagreement at one pair **be** a proof of error? _______________________________

Exit Ticket · Lesson 10.5

  1. 20x6y−25x2y3=\dfrac{20x^6y^{-2}}{5x^2y^3} = _______________________

  2. (3m2n)39mn4=\dfrac{(3m^2n)^3}{9mn^4} = _______________________

  3. 6×1073×10−2=\dfrac{6 \times 10^7}{3 \times 10^{-2}} = ____________ (scientific notation)

  4. Explain. With an outer exponent over a whole ratio, which step comes first, and why does that shorten the work?

PAGE 30 — Chapter 10 review · the derivations

Chapter 10 Review

Part A · Deriving the laws from patterns

FIGURE: fig1-product-rule-expansion.png (half width)

  1. What each brace counts: ______, ______, ______

    The product law: am⋅an=a^m \cdot a^n = ______ Why the exponents add: _______________________

FIGURE: fig2-quotient-rule-cancellation.png (half width)

  1. What the strikethroughs show: _______________________ Each canceled pair is worth: ______

    The quotient law: aman=\dfrac{a^m}{a^n} = ______, for a‾0a \underline{\hspace{1cm}} 0

FIGURE: fig3-quotient-pattern-forces-zero-exponent.png (full width)

  1. Complete the five rows, then the two readings.
Power Value From the row above
242^4
232^3
222^2
212^1
202^0 
 Reading 1: ____________  Reading 2: ______  So a0=a^0 = ______

 Why **forced**, not chosen: _______________________________________

PAGE 31 — Chapter 10 review · the derivations continued

Chapter 10 Review (continued)

FIGURE: fig4-descending-pattern-forces-negative-exponents.png (full width)

  1. Continue the column two more rows.
Power Value As a decimal
2−42^{-4}
2−52^{-5} 
 A negative exponent instructs you to: _______________________________________

 Why 2−12^{-1} cannot be −2-2: _______________________________________

FIGURE: fig5-power-of-a-power-expansion.png (half width)

  1. (am)n=(a^m)^n = ______ The outer exponent counts ____________; the inner counts ____________.

    Why they multiply: _______________________________________

FIGURE: fig6-power-of-a-product-and-quotient.png (full width)

  1. (ab)n=(ab)^n = ____________ (ab)n=\left(\dfrac{a}{b}\right)^n = ____________, for b‾0b \underline{\hspace{1cm}} 0

    Numeric check, left panel: _______________________

    Numeric check, right panel: _______________________

PAGE 32 — Chapter 10 review · reading the exponent

Chapter 10 Review (continued)

FIGURE: fig7-negative-base-versus-negated-power.png (full width)

  1. Give all six values.
Expression Value
(−3)2(-3)^2
−32-3^2
(−2)3(-2)^3
−23-2^3
(−2)4(-2)^4
−24-2^4 
 Rule for reading −an-a^n: _______________________________________

 They agree when: ____________________  They cannot agree when: ____________________

PAGE 33 — Chapter 10 review · multivariable expressions

Chapter 10 Review (continued)

Part B · Multivariable expressions

  1. Simplify, positive exponents.
Expression Answer
a) x5y2⋅x2y6x^5y^2 \cdot x^2y^6
b) −3a4b⋅5a2b3-3a^4b \cdot 5a^2b^3
c) (2m3n2)4(2m^3n^2)^4
d) (p−2q3)−2(p^{-2}q^3)^{-2}

  1. Rewrite with positive exponents only.

    a) 4x−3y=4x^{-3}y = ____________ b) m−2n−5=\dfrac{m^{-2}}{n^{-5}} = ____________

    c) (3ab−2)−1=(3ab^{-2})^{-1} = ____________ d) 2c0d−4=2c^0d^{-4} = ____________

  2. 60=6^0 = ______ 2−5=2^{-5} = ______ (−5)−2=(-5)^{-2} = ______ −5−2=-5^{-2} = ______

    Which two differ only in where a minus sign sits? ____________________

    What that difference does: _______________________________________

PAGE 34 — Chapter 10 review · reasoning about exponents

Chapter 10 Review (continued)

  1. Find the error. A student writes (2x3y)4=2x12y4(2x^3y)^4 = 2x^{12}y^4.

    What went wrong? _______________________ Correct answer: ____________

    Which law was applied to only part of the expression? _______________________

  2. Explain. Why is x−3x^{-3} positive when xx is positive?

    2−32^{-3} exactly: ______ as a decimal: ______

  3. Find each missing exponent.

    a) x7⋅x‾=x3x^7 \cdot x^{\underline{\hspace{2cm}}} = x^3

    b) (a‾)−3=a−15\left(a^{\underline{\hspace{2cm}}}\right)^{-3} = a^{-15}

    c) m−2m‾=m4\dfrac{m^{-2}}{m^{\underline{\hspace{2cm}}}} = m^4

PAGE 35 — Chapter 10 review · ratios of monomials

Chapter 10 Review (continued)

Part C · Ratios of monomial expressions

  1. Simplify, positive exponents.
Ratio Answer
a) x9y4x4y4\dfrac{x^9y^4}{x^4y^4}
b) −32a7b28a3b5\dfrac{-32a^7b^2}{8a^3b^5}
c) (3r2s)26rs3\dfrac{(3r^2s)^2}{6rs^3}
d) 15m−1n625m4n−2\dfrac{15m^{-1}n^6}{25m^4n^{-2}}

  1. 12x5y3z18x2y3z4=\dfrac{12x^5y^3z}{18x^2y^3z^4} = _______________________

    What happened to the base yy, and why? _______________________________________

  2. Check it. Is −32a7b28a3b5=−4a4b3\dfrac{-32a^7b^2}{8a^3b^5} = -\dfrac{4a^4}{b^3}?

Values Original Simplified
a=1a = 1, b=2b = 2
a=2a = 2, b=1b = 1 
 Why is b=0b = 0 not an allowed test value? _______________________________________

  1. Explain. The order for simplifying a ratio of monomials with an outer exponent, and why every step assumes each denominator variable is nonzero.

PAGE 36 — Chapter 10 review · mixed application

Chapter 10 Review (continued)

Part D · Application

FIGURE: fig8-orders-of-magnitude-scale.png (full width)

  1. Apply it. Length marked at 10010^0: ______ Object labeled there: _______________________

    11 km compared with 11 mm: ______ Richmond–Chicago compared with a credit card: ______

    Law used: _______________________

    Why the answer depends on a negative exponent meaning a reciprocal: _______________________

FIGURE: fig9-cell-division-doubling-table.png (full width)

  1. Apply it. Cells at hour 00: ______ Power of 22 recording it: ______

    Cells at hour 66: ______ At hour 1212: ______ as a power, ______________ as a number

    21226=\dfrac{2^{12}}{2^6} = ______ What it means about the two moments: _______________________

  2. Apply it. A drive holds 1.2×10121.2 \times 10^{12} bytes; each recording is 3×1063 \times 10^6 bytes.

    Front numbers: ______ Exponents: ______ Answer: ____________ As a number: ______________

    Law used: _______________________

PAGE 37 — Blank pattern tables

Build the Pattern Yourself

FIGURE: fig10-blank-pattern-tables.png (full page)

Use these two blanks for any pattern exploration a teacher assigns alongside this chapter.

Table A — powers of 33. The first value, 33=273^3 = 27, is filled in for you. Fill the rest of the Value column and, in the third column, write the division that produces each value from the row above. The point is to find that the third column says the same thing on every row, all the way past 303^0 and into 3−13^{-1} and 3−23^{-2} — which is exactly why the zero and negative exponents are not new rules.

Table B — count the factors. For each expression, write out the factors, count them, and rewrite the expression as one power. Say to yourself which law each row is: the first two rows are products, the third is a quotient, and the fourth is a power of a power.

Two reminders for every row you fill in:

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