Lessons from the Math Zone: Exponents

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Click to Start Lesson

Mat h ZoneA

= L

x W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h ZoneA

= L

x W

y = mx + b

3x+ 5 = 14

A = pr 2

Lessons fromLessons fromthethe

Math ZoneMath Zone

ExponentsExponents© Copyright 2006, Don Link

Permission Granted for Educational Use Only

1 2 3 4 5

33 33 33 33 33++ ++ ++ ++

33 5××

33 33 33 33 33++ ++ ++ ++

?

33 6××

33++

= 15= 15

= 18= 18

RepeatedRepeated AdditionAddition

Arithmetic ShortcutsArithmetic ShortcutsMat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

1 2 3 4 5

33 33 33 33 33++ ++ ++ ++

33 5××

33 33 33 33 33++ ++ ++ ++

33 6××

33++

1 2 3 4 5

33 33 33 33 33×× ×× ×× ××

33 5^̂

33 33 33 33 33×× ×× ×× ××

?

33 6^̂

33××

= 15= 15

= 18= 18

= 243= 243

= 729= 729

RepeatedRepeated AdditionAddition RepeatedRepeated MultiplicationMultiplication

““caret”caret”

Arithmetic ShortcutsArithmetic Shortcuts

ANALOGYANALOGY

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

33 33 33 33 33×× ×× ×× ××

33 9^̂

33××

= 19,683= 19,683

33 33×× ×× 33××

Another ExampleAnother Example

^

3^919683.

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

33 33 33 33 33×× ×× ×× ××

33 9^̂

33××

= 19,683= 19,683

33 33×× ×× 33××

22 22 22 22 22×× ×× ×× ××

22 9^̂

22××

= 512= 512

22 22×× ×× 22××

Another ExampleAnother ExampleMat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

BaseBase ExponentExponent^̂

33 99^̂

The ExponentExponent tells us how many copies of the BaseBase to multiplymultiply together.

33 99^̂ Multiply 99 copies of 33 together.

Base = 33

Exponent = 99

TerminologyTerminology

(or Power)(or Power)

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Let’s Practice Let’s Practice withwith Calculators Calculators

3^6 = 729= 729

5^6 = 15,625= 15,625

7^7 = 823,543= 823,543

10^4 = 10,000= 10,000

15^7 = 170,859,375= 170,859,375

2^20= 1,048,576= 1,048,576

20^5 = 3,200,000= 3,200,000

1.5^4 = 5.0625= 5.0625

0.2^4 = 0.0016= 0.0016

10.2^4= 10,824.3216= 10,824.3216

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Let’s Practice Let’s Practice withwith Calculators Calculators

3^6 = 729= 729

5^6 = 15,625= 15,625

7^7 = 823,543= 823,543

10^4 = 10,000= 10,000

15^7 = 170,859,375= 170,859,375

2^20= 1,048,576= 1,048,576

20^5 = 3,200,000= 3,200,000

1.5^4 = 5.0625= 5.0625

0.2^4 = 0.0016= 0.0016

10.2^4= 10,824.3216= 10,824.3216

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Good Job!Good Job!

Let’s Practice Let’s Practice withoutwithout Calculators Calculators

3^2 = 9= 9

5^3 = 125= 125

7^3 = 343= 343

10^3 = 1,000= 1,000

15^1 = 15= 15

2^6 = 64= 64

1^9 = 1= 1

0^4 = 0= 0

3^4 = 81= 81

4^3 = 64= 64

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Let’s Practice Let’s Practice withoutwithout Calculators Calculators

3^2 = 9= 9

5^3 = 125= 125

7^3 = 343= 343

10^3 = 1,000= 1,000

15^1 = 15= 15

2^6 = 64= 64

1^9 = 1= 1

0^4 = 0= 0

3^4 = 81= 81

4^3 = 64= 64

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Excellent!Excellent!

FindingFinding the Correct Exponent the Correct Exponent

5×5×5×5×5×5= 5^__= 5^__6611 22 33 44 55 66

8×8×8×8= 8^__= 8^__44

2×2×2×2×2×2×2= 2^__= 2^__77

7×7 = 7^__= 7^__22

1.5×1.5×1.5×1.5×1.5= 1.5^__= 1.5^__55

4×4×4×4× 4×4×4×4× 4×4×4= 4^__= 4^__1111

BaseBase

ExponentExponent

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

FindingFinding the Correct Exponent the Correct Exponent

5×5×5×5×5×5= 5^__= 5^__6611 22 33 44 55 66

8×8×8×8= 8^__= 8^__44

2×2×2×2×2×2×2= 2^__= 2^__77

7×7 = 7^__= 7^__22

1.5×1.5×1.5×1.5×1.5= 1.5^__= 1.5^__55

4×4×4×4×4×4×4×4×4×4×4×4= 4^__= 4^__1313

BaseBase

ExponentExponent

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Cookin’!Cookin’!

933 933 ^̂

““ThreeThree to the ninthninth power”

554 ““FiveFive to the fourthfourth power”

772 ““SevenSeven to the secondsecond power”““or SevenSeven squaredsquared”

10103 ““TenTen to the thirdthird power”““or TenTen cubedcubed”

Exponents Exponents withoutwithout the Caret the Caret

Speaking Math

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

554

Let’s Practice Let’s Practice withwith Calculators Calculators

= 625= 625

94 = 6,561= 6,561

49 = 262,144= 262,144

176 = 24,137,569= 24,137,569

312 = 531,441= 531,441

77 = 823,543= 823,543

0.54 = 0.0625= 0.0625

2.54 = 39.0625= 39.0625

122.52 = 15,006.25= 15,006.25

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

554

Let’s Practice Let’s Practice withwith Calculators Calculators

= 625= 625

94 = 6,561= 6,561

49 = 262,144= 262,144

176 = 24,137,569= 24,137,569

312 = 531,441= 531,441

77 = 823,543= 823,543

0.54 = 0.0625= 0.0625

2.54 = 39.0625= 39.0625

122.52 = 15,006.25= 15,006.25

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Yes Indeed!Yes Indeed!

52

Let’s Practice Let’s Practice withoutwithout Calculators Calculators

= 25= 25

93 = 729= 729

44 = 256= 256

172 = 289= 289

31 = 3= 3

74 = 2,401= 2,401

04 = 0= 0

114 = 1= 1

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

xx1= x 1x = 1

0x = 0

52

Let’s Practice Let’s Practice withoutwithout Calculators Calculators

= 25= 25

93 = 729= 729

44 = 256= 256

172 = 289= 289

31 = 3= 3

74 = 2,401= 2,401

04 = 0= 0

114 = 1= 1

27 = 128= 128

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Right On!Right On!

FindingFinding the Correct Exponent the Correct Exponent

5×5×5×5×5= 5= 5 5511 22 33 44 55

8×8×8×8= 8= 8 44

2×2×2×2×2×2= 2= 2 66

6×6 = 6= 6 22

1.5×1.5×1.5×1.5×1.5= 1.5= 1.5 55

4×4×4×4× 4×4×4×4× 4×4×4×4= 4= 41212

BaseBase

ExponentExponent

??

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

FindingFinding the Correct Exponent the Correct Exponent

5×5×5×5×5= 5= 5 5511 22 33 44 55

8×8×8×8= 8= 8 44

2×2×2×2×2×2= 2= 2 66

6×6 = 6= 6 22

1.5×1.5×1.5×1.5×1.5= 1.5= 1.5 55

4×4×4×4×4×4×4×4×4×4×4= 4= 41212

BaseBase

ExponentExponent

??

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

High Five!High Five!

11

What if the exponent is zero?What if the exponent is zero?

330334 = 81333 = 27332 = 9331 = 3330 =

Let’s Follow a PatternLet’s Follow a Pattern

=–1

–1

÷3

÷3

–1÷3

–1÷3

11

xx0= 1

CLICK for EXTENSION: CLICK for EXTENSION: NegativeNegative Exponents Exponents

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

CLICK for EXTENSION: CLICK for EXTENSION: 0000

?

Name?Name?BaseBase

334

Exponents: Summary and ReviewExponents: Summary and Review

Name?Name?ExponentExponent (or Power)(or Power)

33 99^̂ On calculatorOn calculator

Name?Name?CaretCaret

334= = 33××33××33××33

334“Three to the fourth powerfourth power”

333

332“Three cubedcubed”

“Three squaredsquared”

xx0= 1

xx1= x

The ExponentExponent tells us how many copies of the BaseBase to multiplymultiply together.

Speaking Math

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

For Printing

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Extension: Negative ExponentsExtension: Negative Exponents

332 = 9331 = 3330 = 1

Let’s Extend the PatternLet’s Extend the Pattern

–1 ÷3

–1 ÷3

–1 ÷3

33-1 = 1/3

33-2–1

÷3

= 1/9

=xx -1x1

xx –nxxn1=

=xx -1x1

xx –nxxn1=

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

(-)

5–2

Let’s PracticeLet’s Practice

= 0.04 = 0.04 (1/25)(1/25)

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

With CalculatorsWith Calculators

Use theUse the(-) Key(-) Key

2־ ^50.04

= 0.125 = 0.125 (1/8)(1/8)

5–2

Let’s PracticeLet’s Practice

= 0.04 = 0.04 (1/25)(1/25)

2–3

10–5= 0.00001= 0.00001

0.05–6 = 64,000,000= 64,000,000

3–3= 0.037= 0.037

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

With CalculatorsWith Calculators

037037…037037…

““A A repeating decimalrepeating decimal”

Speaking Math

= 0.125 = 0.125 (1/8)(1/8)

5–2

Let’s PracticeLet’s Practice

= 0.04 = 0.04 (1/25)(1/25)

2–3

10–5= 0.00001= 0.00001

0.05–6 = 64,000,000= 64,000,000

3–3= 0.037= 0.037

1–4 = 1= 1

1 = 1= 1

2–1 = 1/2= 1/2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

With CalculatorsWith Calculators No CalculatorsNo Calculators

037037…037037…

10–5 = 0.00001= 0.00001

4–2 = 1/16= 1/16

10–3 = 1/1000= 1/1000

3–3 = 1/27= 1/27

–12

5–2

Let’s PracticeLet’s Practice

= 0.04 = 0.04 (1/25)(1/25)

2–3 = 0.125 = 0.125 (1/8)(1/8)

10–5= 0.00001= 0.00001

0.05–6 = 64,000,000= 64,000,000

3–3= 0.037= 0.037

1–4 = 1= 1

1–14= 1= 1

2–1 = 1/2= 1/2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

With CalculatorsWith Calculators No CalculatorsNo Calculators

037037…037037…

10–5 = 0.00001= 0.00001

4–2 = 1/16= 1/16

10–3 = 1/1000= 1/1000

3–3 = 1/27= 1/27

Fantastic!Fantastic!Click to RETURN to Main LessonClick to RETURN to Main Lesson

Extension: Zero to the Zero Power?Extension: Zero to the Zero Power?Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

0000

?? What does this mean?What does this mean?

Rule 1: Rule 1: xx00= 1= 1

Rule 2: Rule 2: = 0= 000xx

Anything to the 0 power = 1.Anything to the 0 power = 1.

Zero to any power = 0.Zero to any power = 0.

Which rule should we use?Which rule should we use?

Extension: Zero to the Zero Power?Extension: Zero to the Zero Power?Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

Mat h Zone

A =

L x

W

y = mx + b

3x+ 5 = 14

A = pr 2

0000

?? What does this mean?What does this mean?

When mathematicians have two perfectly When mathematicians have two perfectly good rules that give different answers for good rules that give different answers for

some problem like some problem like 0000, they say the answer , they say the answer is __________ for this case.is __________ for this case.

undefinedundefinedSo, is undefined!So, is undefined!0000

Click to RETURN to Main LessonClick to RETURN to Main Lesson