Two pages of visual notes for students to keep: the key words with each rule written out the long way, the pattern that explains zero and negative exponents, four solved problems in the style of the Exponent Racer game, and where exponents show up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Exponents and the Exponent Rules
Grades 6–8 · CCSS 6.EE.A.1, 8.EE.A.1 · Pairs with the Exponent Racer game
Name
The big idea: an exponent says how many times to use a number as a factor. 2³ = 2 × 2 × 2 = 8. Every exponent rule comes from writing the factors out and counting them. Forget a rule? Write it out the long way and count.
Key vocabulary
Base
43
The number being multiplied: 4.
Exponent
43
How many times the base is a factor: 3.
Power
4³ = 4·4·4 = 64
A base with an exponent, and its value.
Product of powers
2³·2² = (2·2·2)(2·2) = 2⁵
Same base, multiply → add the exponents.
Quotient of powers
5⁴ ÷ 5² = 5·5·5·5⁄5·5 = 5²
Same base, divide → subtract the exponents.
Power of a power
(3²)³ = 3²·3²·3² = 3⁶
Multiply the exponents.
Zero exponent
7⁰ = 1
Any nonzero base to the 0 power is 1.
Negative exponent
5⁻² = 1⁄5² = 1⁄25
One over the positive power.
Why a⁰ = 1 and negatives make fractions: follow the pattern
2³
2²
2¹
2⁰
2⁻¹
2⁻²
8
4
2
1
½
¼
← each step to the right: ÷ 2 ←
Each time the exponent goes down by 1, the value is divided by 2. Keep going past 2¹ = 2: the next value is 1, so 2⁰ = 1. Then ½, then ¼: negative exponents make fractions, not negative numbers.
Same base (5)? Yes, so add the exponents: 2 + 3 = 5
5² × 5³ = 5⁵
Long way: (5·5)(5·5·5): five 5s
✓ 5⁵ = 3,125
(2²)³ = ?
3. Power of a power
Multiply the exponents: 2 × 3 = 6
(2²)³ = 2⁶ = 64
Long way: 2² · 2² · 2² = 4 · 4 · 4 = 64
✓ Not 2⁵: that would be adding
4³ ÷ 4⁵ = ?
4. A negative exponent
Divide → subtract: 3 − 5 = −2, so 4⁻²
Make it positive: 4⁻² = 1⁄4² = 1⁄16
✓ Long way: two 4s are left on the bottom
Watch out
2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.
The rules only work with the same base: 2³ × 5² can't be combined into one power.
A negative exponent makes a fraction, not a negative number: 2⁻³ = ⅛.
Where you see it in real life
Cells and bacteria double: 1, 2, 4, 8 = 2⁰, 2¹, 2², 2³.Space: the Sun is about 1.5 × 10⁸ km away.Floors: a 4-by-4 square of tiles is 4² = 16 tiles.Boxes: a cube's volume is its edge cubed, like 3³.