Step by Step
A
Add — same base, multiplication
When multiplying powers with the same base, add the exponents: xᵃ · xᵇ = x^(a+b).
M
Subtract — same base, division
When dividing powers with the same base, subtract the exponents: xᵃ / xᵇ = x^(a−b).
S
Multiply — power of a power
When raising a power to another power, multiply the exponents: (xᵃ)ᵇ = x^(a×b).
Z
Zero — the zero exponent rule
Any nonzero base raised to the zero power equals 1: x⁰ = 1, not 0.
Applied Walkthrough
1
Simplify x³ · x⁴. Same base, multiplying — add the exponents: x³ · x⁴ = x⁷.
2
Now compare that to (x³)⁴ — this is a power raised to a power, so you multiply the exponents instead: (x³)⁴ = x¹².
3
These look similar but use completely different rules — mixing them up (getting x¹² for the first, or x⁷ for the second) is the most common exponent mistake.
4
Finally, note that x³ · y⁴ cannot be simplified at all using AMSZ, since the bases (x and y) don't match — these rules only apply when the base is the same.
Exam Application
Exams test whether you can correctly distinguish adding exponents (same-base multiplication) from multiplying exponents (power of a power), and whether you know the zero exponent rule and the same-base requirement.
⚠ Common Trap
The most common trap is confusing the Add rule with the Multiply rule — x³ · x⁴ (add exponents, same-base multiplication) looks deceptively similar to (x³)⁴ (multiply exponents, power of a power), but they use different rules entirely.
✓ Quick Self-Check
1. What does AMSZ stand for?
Add, Multiply, Subtract, Zero (the four core exponent rules).
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2. Simplify x³ · x⁴.
x⁷ (add the exponents, since it's same-base multiplication).
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3. Simplify (x³)⁴.
x¹² (multiply the exponents, since it's a power raised to a power).
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4. What does x⁰ equal?
1, for any nonzero base — not 0.
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5. Can x³ · y⁴ be simplified using AMSZ?
No — the bases must match for these exponent rules to apply.
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