Exponents & Polynomials
Negative & Zero Exponents
Simplifying expressions with negative and zero exponents.
Prerequisites
- Exponent Rules
Rewriting an Expression with Only Positive Exponents
Before reading on, try simplifying x⁻³y² / (x²y⁻¹) using what you already know: any base to the power 0 equals 1, and a⁻ⁿ = 1/aⁿ. Work through each base, x and y, separately.
Handle one base at a time. For x: the expression has x⁻³ in the numerator and x² in the denominator, which by the quotient rule combines to x⁻³⁻² = x⁻⁵. For y: y² in the numerator and y⁻¹ in the denominator combines to y²⁻(⁻¹) = y³. Putting both together: x⁻⁵y³, which rewritten with only positive exponents is y³/x⁵.
Worked Example — Simplifying with Negative Exponents
Worked Example — Simplifying an Expression with a Zero Exponent
Tip
Common Mistakes
Treating a negative exponent as making the entire term negative, such as simplifying x⁻² as −x².
A negative exponent signals a reciprocal, not a negative value — x⁻² = 1/x², which stays positive whenever x is positive.
Combining exponents from two different bases as if they were the same, such as adding an x exponent to a y exponent.
Only exponents on the exact same base combine using the product or quotient rules — x's exponents and y's exponents must be tracked and simplified completely separately.
Key Takeaways
- A negative exponent rewrites as a reciprocal with a positive exponent: a⁻ⁿ = 1/aⁿ.
- Any nonzero base raised to the power 0 equals exactly 1.
- In an expression with multiple bases, simplify each base's exponents independently before combining the results.
Summary
With the full exponent toolkit in place, the next unit uses it to work with polynomials — starting with adding and subtracting them.
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