Lessons from the Math Zone: Exponents Click to Start Lesson.

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Presentation transcript:

Lessons from the Math Zone: Exponents Click to Start Lesson

Lessons from the Math Zone Exponents © Copyright 2006, Don Link Permission Granted for Educational Use Only

× ? 36× 3+ = 15 = 18 Repeated Addition Arithmetic Shortcuts

× × ×××× 35^ 33333×××× ? 36^ 3× = 15 = 18 = 243 = 729 Repeated Addition Repeated Multiplication “caret” Arithmetic Shortcuts ANALOGY

33333×××× 39^ 3× = 19,683 33××3× Another Example ^ 3^

33333×××× 39^ 3× = 19,683 33××3× 22222×××× 29^ 2× = ××2× Another Example

BaseExponent^ 39^ The E EE Exponent tells us how many copies of the B BB Base to m mm multiply together. 39^ Multiply copies of together. 3 Base = 3 9 Exponent = 9 Terminology (or Power)

Let’s Practice with Calculators 3^63^6 = 729 5^65^6 = 15,625 7^77^7 = 823, ^4 = 10, ^7 = 170,859,375 2 ^20 = 1,048, ^5 = 3,200, ^4 = ^4 = ^4 = 10,

Let’s Practice with Calculators 3^63^6 = 729 5^65^6 = 15,625 7^77^7 = 823, ^4 = 10, ^7 = 170,859,375 2 ^20 = 1,048, ^5 = 3,200, ^4 = ^4 = ^4 = 10, Good Job!

Let’s Practice without Calculators 3^23^2 = 9 5^35^3 = 125 7^37^3 = ^3 = 1, ^1 = 15 2^62^6 = 64 1^91^9 = 1 0^40^4 = 0 3^43^4 = 81 4^34^3 = 64

Let’s Practice without Calculators 3^23^2 = 9 5^35^3 = 125 7^37^3 = ^3 = 1, ^1 = 15 2^62^6 = 64 1^91^9 = 1 0^40^4 = 0 3^43^4 = 81 4^34^3 = 64 Excellent!

Finding the Correct Exponent 5×5×5×5×5×55×5×5×5×5×5 = 5^__ ×8×8×88×8×8×8 = 8^__ 4 2×2×2×2×2×2×22×2×2×2×2×2×2 = 2^__ 7 7×77×7 = 7^__ × 1.5 × 1.5 × 1.5 × 1.5 = 1.5^__ 5 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 4^__ 11 Base Exponent

Finding the Correct Exponent 5×5×5×5×5×55×5×5×5×5×5 = 5^__ ×8×8×88×8×8×8 = 8^__ 4 2×2×2×2×2×2×22×2×2×2×2×2×2 = 2^__ 7 7×77×7 = 7^__ × 1.5 × 1.5 × 1.5 × 1.5 = 1.5^__ 5 4×4×4×4×4×4×4×4×4×4×4×44×4×4×4×4×4×4×4×4×4×4×4 = 4^__ 13 Base Exponent Cookin’!

9393^ “Three to the n nn ninth power” 5 4 “Five to the f ff fourth power” 7 2 “Seven to the s ss second power” “or S SS Seven s ss squared” 10 3 “Ten to the t tt third power” “or T TT Ten c cc cubed” Exponents without the Caret

5 4 Let’s Practice with Calculators = = 6, = 262, = 24,137, = 531, = 823, = = = 15,006.25

5 4 Let’s Practice with Calculators = = 6, = 262, = 24,137, = 531, = 823, = = = 15, Yes Indeed!

5 2 Let’s Practice without Calculators = = = = = = 2, = = 1 0 x =0

5 2 Let’s Practice without Calculators = = = = = = 2, = = = 128 Right On!

Finding the Correct Exponent 5×5×5×5×55×5×5×5×5 = ×8×8×88×8×8×8 = 8 4 2×2×2×2×2×22×2×2×2×2×2 = 2 6 6×66×6 = × 1.5 × 1.5 × 1.5 × 1.5 = × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 4 12 Base Exponent ?

Finding the Correct Exponent 5×5×5×5×55×5×5×5×5 = ×8×8×88×8×8×8 = 8 4 2×2×2×2×2×22×2×2×2×2×2 = 2 6 6×66×6 = × 1.5 × 1.5 × 1.5 × 1.5 = ×4×4×4×4×4×4×4×4×4×44×4×4×4×4×4×4×4×4×4×4 = 4 12 Base Exponent ? High Five!

1 What if the exponent is zero? = 81 = 27 = 9 = 3 = Let’s Follow a Pattern = –1–1 –1–1 ÷3÷3 ÷3÷3 –1–1 ÷3÷3 –1–1 ÷3÷3 1 = 1 CLICK for EXTENSION: Negative Exponents CLICK for EXTENSION: Negative Exponents CLICK for EXTENSION: 0 CLICK for EXTENSION: 0 ?

Name? Base Exponents: Summary and Review Name? Exponent (or Power) On calculator Name? Caret = 3× 3×3×3 fourth power “Three to the fourth power” cubed “Three cubed” squared “Three squared”

For Printing

Extension: Negative Exponents = 9 = 3 = 1 Let’s Extend the Pattern –1–1 ÷3÷3 –1–1 ÷3÷3 –1–1 ÷3÷3 = 1/3 –1–1 ÷3÷3 = 1/9

(-) 5 –2–2 Let’s Practice = 0.04 (1/25) With Calculators Use the (-) Key 5^ ־2 0.04

= (1/8) 5 –2–2 Let’s Practice = 0.04 (1/25) 2 –3–3 10 –5–5 = –6–6 = 64,000,000 3 –3–3 = With Calculators … “A repeating decimal”

= (1/8) 5 –2–2 Let’s Practice = 0.04 (1/25) 2 –3–3 10 –5–5 = –6–6 = 64,000,000 3 –3–3 = –4–4 = –1–1 = 1/2 With Calculators No Calculators … 10 –5–5 = –2–2 = 1/16 10 –3–3 = 1/ –3–3 = 1/27 –12

5 –2–2 Let’s Practice = 0.04 (1/25) 2 –3–3 = (1/8) 10 –5–5 = –6–6 = 64,000,000 3 –3–3 = –4–4 = 1 1 –14 = 1 2 –1–1 = 1/2 With Calculators No Calculators … 10 –5–5 = –2–2 = 1/16 10 –3–3 = 1/ –3–3 = 1/27 Fantastic! Click to RETURN to Main Lesson Click to RETURN to Main Lesson

Extension: Zero to the Zero Power? 0 0 ? What does this mean? Rule 1: x 0 = 1 Rule 2: = 0 0 x Anything to the 0 power = 1. Zero to any power = 0. Which rule should we use?

Extension: Zero to the Zero Power? 0 0 ? What does this mean? When mathematicians have two perfectly good rules that give different answers for some problem like 00, they say the answer is __________ for this case. undefined So, is undefined! 00 Click to RETURN to Main Lesson Click to RETURN to Main Lesson