Same Math › Grade 8 › Exponent rules by expanding
How are exponent rules taught now?
Instead of memorizing rules first, kids write exponents out as repeated multiplication: x³ · x² is (x·x·x)(x·x), five x's, so x⁵. After a few, they notice the rule themselves: when you multiply, add the exponents.
- Expand: write out every factor.
- Count: how many are multiplied?
- The rule (add exponents) is a shortcut for counting.
Why schools teach it
Rules you can rebuild don't get mixed up
Eighth graders know and apply the properties of integer exponents, including zero and negative exponents (8.EE.A.1). The rules look alike, so kids confuse when to add, subtract or multiply exponents.
Expanding settles it every time. x⁵ ÷ x² cancels two x's, leaving x³: subtract. (x²)³ is three groups of two: multiply.
At the kitchen table
What to say, and what to skip
Say this
Write it out. How many x's in all?
Which x's cancel when you divide?
So what's the shortcut?
Let them name the rule.What's 2³? Write it out first.
2 × 2 × 2, not 2 × 3.
Skip this
2³ is 6.
It's 2 × 2 × 2 = 8.- Memorizing the rules list first.Expand a few, then the rules make sense.
Anything to the zero is zero.
It's 1. x³ ÷ x³ = 1, and the rule says x⁰.
See it in action
Why x⁰ = 1
Divide something by itself
x³ ÷ x³ = 1Use the subtract rule
x³ ÷ x³ = x³⁻³ = x⁰Both are true, so
x⁰ = 1
The same reasoning shows x⁻² = 1 ÷ x².
Free practice sheet
Make an exponent rules sheet
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Parents also ask
What are the main rules?
Multiply: add exponents. Divide: subtract. Power of a power: multiply. Zero exponent: 1. Negative exponent: one over.
Why is x⁻² equal to 1 ÷ x²?
x³ ÷ x⁵ leaves two x's on the bottom: 1 ÷ x². The rule gives x⁻².
Do the rules work for numbers too?
Yes: 2³ · 2² = 2⁵ = 32.
Where do exponents show up next?
Scientific notation in eighth grade, then polynomials in Algebra 1.