Exponent Rules | Naruhodo Tutoring

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xᵃ · xᵇ = xᵃ⁺ᵇ

x³ · x⁴

Base (x)

Exponent a

Exponent b

Base (x)

Exponent a

Exponent b

Base (x)

Inner exp a

Outer exp b

Base x

Base y

Exponent n

Numerator x

Denominator y

Exponent n

Base (any non-zero)

Base (x)

Exponent n (positive)

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Result

Select a rule, enter your values, then press Simplify to see the result and steps.

Simplified Result

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All 7 Exponent Rules — Quick Reference

Rule	Formula	Example
Product Rule	xᵃ · xᵇ = xᵃ⁺ᵇ	x² · x³ = x⁵
Quotient Rule	xᵃ / xᵇ = xᵃ⁻ᵇ	x⁷ / x³ = x⁴
Power Rule	(xᵃ)ᵇ = xᵃᵇ	(x³)² = x⁶
Power of Product	(xy)ⁿ = xⁿyⁿ	(xy)³ = x³y³
Power of Quotient	(x/y)ⁿ = xⁿ/yⁿ	(x/y)² = x²/y²
Zero Exponent	x⁰ = 1 (x ≠ 0)	5⁰ = 1
Negative Exponent	x⁻ⁿ = 1/xⁿ	x⁻³ = 1/x³

All rules require the same base for product and quotient rules. You can only combine xᵃ · xᵇ if both bases are x — you cannot combine x² · y³ this way.

Common Mistakes to Avoid

x² · x³ ≠ x⁶ (correct: x⁵ — add exponents, don't multiply them)
(x + y)² ≠ x² + y² (correct: expand using FOIL — exponents don't distribute over addition)
x⁰ ≠ 0 (correct: x⁰ = 1 for any non-zero x)
x⁻² ≠ −x² (correct: x⁻² = 1/x² — a negative exponent means a reciprocal)
(2x)³ ≠ 2x³ (correct: (2x)³ = 8x³ — the coefficient is also raised to the power)
x² + x³ cannot be simplified (you can only combine like terms, not different powers)

The product and quotient rules only work when bases are identical. x² · y³ is already fully simplified.

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