Lessons from the Math Zone Exponents.

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Presentation transcript:

Lessons from the Math Zone Exponents

Arithmetic Shortcuts Repeated Addition 3 + 3 + 3 + 3 + 3 1 2 3 4 5 3 × ? 5 = 15 3 × 6 = 18 3 + 3 + 3 + 3 + 3 + 3

Arithmetic Shortcuts Repeated Addition Repeated Multiplication 3 + 3 + 3 + 3 + 3 3 × 3 × 3 × 3 × 3 ANALOGY 1 2 3 4 5 1 2 3 4 5 3 × 5 = 15 3 ^ 5 ? = 243 “caret” 3 × 6 = 18 3 ^ 6 = 729 3 + 3 + 3 + 3 + 3 + 3 3 × 3 × 3 × 3 × 3 × 3

3 ^ 9 = 19,683 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 ^ Another Example 3^9 19683. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 ^

Another Example 3 ^ 9 = 19,683 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 2 ^ 9 = 512 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

3 ^ 9 Base ^ Exponent (or Power) 3 ^ 9 Terminology The Exponent tells us how many copies of the Base to multiply together. Base = 3 3 ^ 9 Multiply 9 copies of 3 together. Exponent = 9

Let’s Practice with Calculators 3^6 = 729 1.5^4 = 5.0625 5^6 = 15,625 0.2^4 = 0.0016 7^7 = 823,543 10.2^4 = 10,824.3216 10^4 = 10,000 15^7 = 170,859,375 2^20 = 1,048,576 20^5 = 3,200,000

Let’s Practice with Calculators 3^6 = 729 1.5^4 = 5.0625 5^6 = 15,625 0.2^4 = 0.0016 7^7 = 823,543 10.2^4 = 10,824.3216 10^4 = 10,000 15^7 = 170,859,375 2^20 = 1,048,576 20^5 = 3,200,000

Let’s Practice without Calculators 3^2 = 9 0^4 = 0 5^3 = 125 3^4 = 81 7^3 = 343 4^3 = 64 10^3 = 1,000 15^1 = 15 2^6 = 64 1^9 = 1

Let’s Practice without Calculators 3^2 = 9 0^4 = 0 5^3 = 125 3^4 = 81 7^3 = 343 4^3 = 64 10^3 = 1,000 15^1 = 15 2^6 = 64 1^9 = 1

Finding the Correct Exponent 1 2 3 4 5 6 5×5×5×5×5×5 = 5^__ 6 Base 8×8×8×8 = 8^__ 4 2×2×2×2×2×2×2 = 2^__ 7 7×7 = 7^__ 2 1.5×1.5×1.5×1.5×1.5 = 1.5^__ 5 4×4×4×4× 4×4×4×4× 4×4×4 = 4^__ 11

Finding the Correct Exponent 1 2 3 4 5 6 5×5×5×5×5×5 = 5^__ 6 Base 8×8×8×8 = 8^__ 4 2×2×2×2×2×2×2 = 2^__ 7 7×7 = 7^__ 2 1.5×1.5×1.5×1.5×1.5 = 1.5^__ 5 4×4×4×4×4×4×4×4×4×4×4×4 = 4^__ 13

Exponents without the Caret 3 3 ^ 9 9 “Three to the ninth power” 4 5 “Five to the fourth power” 2 7 “Seven to the second power” “or Seven squared” 3 10 “Ten to the third power” “or Ten cubed”

Let’s Practice with Calculators 4 4 5 = 625 0.5 = 0.0625 4 4 9 = 6,561 2.5 = 39.0625 9 2 4 = 262,144 122.5 = 15,006.25 6 17 = 24,137,569 12 3 = 531,441 7 7 = 823,543

Let’s Practice with Calculators 4 4 5 = 625 0.5 = 0.0625 4 4 9 = 6,561 2.5 = 39.0625 9 2 4 = 262,144 122.5 = 15,006.25 6 17 = 24,137,569 12 3 = 531,441 7 7 = 823,543

Let’s Practice without Calculators 2 4 5 = 25 = 0 3 x = 9 = 729 4 4 = 256 14 1 = 1 2 17 = 289 1 3 = 3 4 7 = 2,401

Let’s Practice without Calculators 2 4 5 = 25 = 0 3 14 9 = 729 1 = 1 4 7 4 = 256 2 = 128 2 17 = 289 1 3 = 3 4 7 = 2,401

Finding the Correct Exponent 1 2 3 4 5 5×5×5×5×5 = 5 5 ? Base 8×8×8×8 = 8 4 6 2×2×2×2×2×2 = 2 2 6×6 = 6 5 1.5×1.5×1.5×1.5×1.5 = 1.5 12 4×4×4×4× 4×4×4×4× 4×4×4×4 = 4

Finding the Correct Exponent 1 2 3 4 5 5×5×5×5×5 = 5 5 ? Base 8×8×8×8 = 8 4 6 2×2×2×2×2×2 = 2 2 6×6 = 6 5 1.5×1.5×1.5×1.5×1.5 = 1.5 12 4×4×4×4×4×4×4×4×4×4×4 = 4

What if the exponent is zero? CLICK for EXTENSION: Negative Exponents Let’s Follow a Pattern = 81 = ? –1 ÷3 = 27 –1 ÷3 = 1 = 9 –1 ÷3 = 3 –1 ÷3 = 1 1 CLICK for EXTENSION: Negative Exponents CLICK for EXTENSION: 00

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

For Printing

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

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

Let’s Practice With Calculators –2 5 = 0.04 (1/25) –3 2 = 0.125 (1/8) –5 10 = 0.00001 –6 0.05 = 64,000,000 –3 3 = 0.037 037037… “A repeating decimal”

Let’s Practice With Calculators No Calculators –2 –4 5 = 0.04 (1/25) 1 = 1 –3 –12 2 = 0.125 (1/8) 1 = 1 –5 –1 10 = 0.00001 2 = 1/2 –6 –2 0.05 = 64,000,000 4 = 1/16 –3 3 –3 = 0.037 037037… 3 = 1/27 –5 10 –3 = 0.00001 10 = 1/1000

Let’s Practice With Calculators No Calculators –2 –4 5 = 0.04 (1/25) 1 = 1 –3 –14 2 = 0.125 (1/8) 1 = 1 –5 –1 10 = 0.00001 2 = 1/2 –6 –2 0.05 = 64,000,000 4 = 1/16 –3 3 –3 = 0.037 037037… 3 = 1/27 –5 10 –3 = 0.00001 10 = 1/1000

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

Extension: Zero to the Zero Power? Click to RETURN to Main Lesson ? 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. So, is undefined! undefined Click to RETURN to Main Lesson