Monday
Monday: Powers That Multiply
Rocket is stringing lights across the Puzzle Post. "Puzzle Night is Saturday. I need a huge-number puzzle."
He folds a sheet in half three times, then another three times. "2³ layers times 2³ layers. That is 8 × 8 = 64."
"What do you notice about 64 as a power of 2?" Raven asks.
"It is 2⁶," Rocket says. "Three factors of 2, then three more. The exponents added."
Nova hovers over the folded sheets, her light counting the layers. "And if you divide 2⁶ by 2²?"
"Six factors, take away two. 2⁴, which is 16," Rocket says. "Subtract the exponents."
Raven writes 2⁰ on a card. "And this?"
"2² ÷ 2² is 1, and that is 2⁰. So anything to the zero is 1."
Three rules for one base
A power is repeated multiplication. 2³ means 2 × 2 × 2 = 8. The exponent counts the factors.
Multiply powers of the same base and the factors pile up. 2³ × 2³ has 6 factors of 2, so it is 2⁶ = 64.
Divide powers of the same base and factors cancel: 2⁶ ÷ 2² leaves 4 factors, 2⁴ = 16.
Divide a power by itself and you get 1. That is why 2⁰ = 1, and 7⁰ = 1 too.
| Rule | In symbols | Example |
|---|---|---|
| Multiply: add the exponents | Count all the factors together | 3² × 3⁵ = 3⁷ |
| Divide: subtract the exponents | Matching factors cancel | 3⁵ ÷ 3² = 3³ |
| Zero exponent | A power divided by itself is 1 | 3⁰ = 1 |
| Negative exponent | One divided by the positive power | 3⁻² = 1 ÷ 3² = 1/9 |
Negative exponents
Keep dividing by 3: 3² = 9, 3¹ = 3, 3⁰ = 1. The next step down is 1/3, written 3⁻¹.
So a negative exponent means one divided by the positive power. 3⁻² = 1 ÷ 9 = 1/9, and 2⁻³ = 1/8.
The rules still hold: 3² × 3⁻⁵ = 3⁻³ = 1 ÷ 27, because 2 + (-5) = -3.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| 2³ × 2³ = 2⁶ | ? |
| 2³ × 2³ = 2⁹ | ? |
| 5⁶ ÷ 5² = 5⁴ | ? |
| 3⁻² is a negative number. | ? |
| Any base to the power 0 is 1. | ? |
Stack the two folded sheets. Write the total layers as a power of 2 and check the exponents added.
Strong start. Tomorrow you undo squares and cubes with beans in a square and blocks in a cube.