Exponent Rules Calculator
Apply product, quotient, and power rules for exponents with step-by-step solutions
Understanding Exponent Rules
Exponent rules (also called laws of exponents) are mathematical properties that make working with powers easier. These rules apply when bases are the same.
The Three Main Exponent Rules
1. Product Rule
a^m × a^n = a^(m+n)
When multiplying powers with the same base, add the exponents.
Example: x³ × x² = x^(3+2) = x⁵
2. Quotient Rule
a^m ÷ a^n = a^(m-n)
When dividing powers with the same base, subtract the exponents.
Example: x⁵ ÷ x² = x^(5-2) = x³
3. Power Rule
(a^m)^n = a^(m×n)
When raising a power to a power, multiply the exponents.
Example: (x³)² = x^(3×2) = x⁶
Additional Exponent Rules
- Zero Exponent: a⁰ = 1 (for a ≠ 0)
- Negative Exponent: a^(-n) = 1/a^n
- Power of a Product: (ab)^n = a^n × b^n
- Power of a Quotient: (a/b)^n = a^n / b^n
Important Notes
- These rules only work when the bases are the same
- You cannot use these rules to simplify x² + x³ (different operations)
- Be careful with negative exponents - they create fractions
- Always simplify exponents before evaluating if possible
Frequently Asked Questions
When can I use exponent rules?
Exponent rules can only be applied when the bases are the same. For example, you can use the product rule on x³ × x², but not on x³ × y².
Why do we add exponents when multiplying?
Because multiplication is repeated addition. x³ × x² means (x·x·x) × (x·x) = x·x·x·x·x = x⁵. We have 3 + 2 = 5 total factors of x.
What happens when the quotient rule gives a negative exponent?
A negative exponent means the result is a fraction. For example, x² ÷ x⁵ = x^(2-5) = x^(-3) = 1/x³. The result "flips" to the denominator.
Can I use the product rule on x² + x³?
No! The product rule only works for multiplication, not addition. x² + x³ cannot be simplified using exponent rules because the terms are being added, not multiplied.
How do I remember which operation goes with which rule?
Think about it logically: multiplying means you're adding more factors (add exponents), dividing means you're taking factors away (subtract exponents), and raising to a power means repeated multiplication (multiply exponents).
Do exponent rules work with variables in the exponent?
Yes! The rules work the same way. For example, x^a × x^b = x^(a+b), even though a and b are variables. The base must still be the same.
What if the bases are different but the exponents are the same?
Use the power of a product rule: x^n × y^n = (xy)^n. For example, 2³ × 5³ = (2×5)³ = 10³ = 1000.
Can fractions be bases?
Yes! All exponent rules work with fractional bases. For example, (1/2)³ × (1/2)² = (1/2)⁵. Just make sure the bases are identical.
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