Exponent Rules Calculator 2025: Complete Guide
Exponent rules are the foundation of algebra and scientific notation. Our Exponent Rules Calculator applies all 7 core rules instantly with step-by-step explanations. Whether you're simplifying (x^2)^3 or converting x^-2, master exponents in seconds.
The 7 Core Exponent Rules
1. Product Rule: a^m × a^n = a^(m+n)
Example: x^3 × x^5 = x^8
2. Quotient Rule: a^m ÷ a^n = a^(m-n)
Example: x^7 ÷ x^2 = x^5
3. Power of Power: (a^m)^n = a^(m×n)
Example: (x^2)^4 = x^8
4. Power of Product: (ab)^n = a^n × b^n
Example: (2x)^3 = 8x^3
5. Negative Exponent: a^-n = 1/a^n
Example: x^-3 = 1/x^3
6. Zero Exponent: a^0 = 1 (a ≠ 0)
Example: 5^0 = 1
7. Fractional Exponent: a^(p/q) = (q-th root of a)^p
Example: x^(1/2) = sqrt(x)
Step-by-Step: Simplify (x^2)^3 × x^4
- Power Rule:
(x^2)^3 = x^(2×3) = x^6 - Product Rule:
x^6 × x^4 = x^(6+4) = x^10 - Final:
x^10
Negative Exponents in Fractions
1/x^-2 = x^2 (move to denominator)
Common Mistakes to Avoid
(x + y)^2 ≠ x^2 + y^2(use binomial theorem)x^2 + x^3 ≠ x^5(different exponents)- Forgetting
a^0 = 1 - Misapplying negative exponents in products
Advanced Applications
- Scientific Notation:
3.2 × 10^8 - Logarithms:
log(x^3) = 3log(x) - Binomial Theorem:
(a + b)^n
Real-World Uses
Exponents model:
- Compound interest:
A = P(1 + r)^t - Population growth:
P = P_0 × e^(rt) - Radioactive decay:
N = N_0 × (1/2)^t - Signal strength:
dB = 10log(P/P_0)
FAQs
Can I use variables other than x? Yes — y^3, a^-2, etc.
What about (2x)^3? Expands to 8x^3 using power of product.
Does it handle roots? Yes — x^(1/3) = cube root of x.
Conclusion
Exponent rules are essential for algebra, calculus, and science. Our 2025 Exponent Rules Calculator eliminates confusion with instant, accurate, step-by-step solutions. From homework to real-world modeling, simplify exponents confidently. Enter your expression now!
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