Two pages of visual notes for students to keep: the key words with each rule written out the long way, the rules at a glance, four solved problems in the style of the Exponent Hoops game, and where exponents show up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Exponent Rules with Variables
Grade 8, Algebra 1 · CCSS 8.EE.A.1 · Pairs with the Exponent Hoops game
Name
The big idea: x⁵ means x · x · x · x · x: five x's multiplied. The exponent rules are just counting x's. Multiply and the x's pile up (add). Divide and they cancel (subtract). Raise a power to a power and you copy the group (multiply).
Key vocabulary
Base
x5
The number or variable being multiplied: x.
Exponent
x5
How many times the base is a factor: 5.
Power
x³ = x·x·x
A base with an exponent.
Product of powers
x²·x³ = (x·x)(x·x·x) = x⁵
Same base, multiply: add exponents.
Quotient of powers
x⁵ ÷ x² = x·x·x·x·x⁄x·x = x³
Same base, divide: subtract exponents.
Power of a power
(x²)³ = x²·x²·x² = x⁶
Multiply the exponents.
Zero exponent
x⁰ = 1
x³ ÷ x³ = 1, and also x⁰. Any nonzero base to the 0 is 1.
Visual Notes: Exponent Rules with Variables, page 2
m⁴ · m³
1. Multiply
Same base? Yes, m. Multiplying, so add: 4 + 3 = 7
m⁷
✓ Count: (m·m·m·m)(m·m·m) is seven m's
k⁹ ÷ k⁴
2. Divide
Same base? Yes, k. Dividing, so subtract: 9 − 4 = 5
k⁵
✓ Four k's cancel from top and bottom, five are left
(p³)⁴
3. Power of a power
Four groups of p³: p³ · p³ · p³ · p³
Multiply: 3 × 4 = 12, so p¹²
✓ Not p⁷: that would be adding
b² · b³ ÷ b⁷
4. Put the rules together
Multiply first: b² · b³ = b⁵
Then divide: b⁵ ÷ b⁷ = b⁻²
Positive exponent: b⁻² = 1⁄b²
✓ Five b's on top, seven on the bottom: two are left below
Watch out
x² means x · x, not 2x. Test with x = 3: 9 is not 6.
Add exponents when you multiply powers; multiply exponents for a power of a power.
The rules need the same base: x² · y³ stays as it is.
Where you see it in real life
Growth: doubling each hour gives 2ⁿ after n hours.Area: a square with side s has area s².Volume: a cube with edge s has volume s³.Science: huge numbers are written with powers of 10.