Exponent Rules by Counting

Lesson 2 of 11 · 7 min · Free, no account needed

What you will learn

Multiplying powers of the same base joins the rows: 2ᵃ · 2ᵇ = 2ᵃ⁺ᵇ.
A power of a power stamps copies of a row: (2ᵃ)ᵇ = 2ᵃᵇ.
Dividing cancels chips in pairs: 2ᵃ / 2ᵇ = 2ᵃ⁻ᵇ. Adding powers follows none of these rules.

2⁵ × 2⁶, no calculator. Can you beat your phone?
The answer is a power of 2. Drag the lever to bet its exponent.
Write the powers as rows of twos and join them.
2⁵ is five twos multiplied. Drag the orange row onto the end of the blue one.
Build (2³)⁴ with a stamp.
The outside exponent says how many copies of the row to print.
A power of a power repeats the whole row. Four copies of a row of 3 hold 3 × 4 chips.
Three rules, all from counting chips.
Multiply: aᵐ · aⁿ = aᵐ⁺ⁿ. The rows join, so the counts add.
Power of a power: (aᵐ)ⁿ = aᵐⁿ. n copies of a row of m.
Divide: aᵐ / aⁿ = aᵐ⁻ⁿ. Each bottom chip cancels a top chip.
Divide by cancelling pairs.
A 2 on top and a 2 on the bottom make 2 ÷ 2 = 1. Tap the chips to cancel them.
Game: the power sorter.
Drop each card into the power it equals. Three lives.
Make 2¹² your own way.
Pick an operation and two exponents so the result is 2¹².
Every rule turns the two exponents into one number: add for ×, subtract for ÷, multiply for a power. You picked two that land on 12.
Computers count in bits.
One bit is 0 or 1: 2 values. Each extra bit doubles the count. How many values fit in two bytes?
Grade Pip's homework.
Tap every wrong line. Your own line from before is in there too.
Rows join only when you multiply. (2³)² stamps two rows of 3: 2⁶. And 2³ + 2³ is two copies of 8, which is 16 = 2⁴, not 2⁶.

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