Multiplying Exponents Calculator
Calculate products of exponential expressions with step-by-step solutions and exponent rules
Calculate x^a × y^b
First Term: x^a
First Term:
1 = undefined
Second Term: y^b
Second Term:
1 = undefined
Advanced Options
⚠️ Invalid calculation: Check for division by zero or undefined operations (e.g., 0^0).
Exponent Rules
Product of Powers
x^a × x^b = x^(a+b)
Same base: add exponents
Power of Product
x^a × y^a = (x×y)^a
Same exponent: multiply bases
Power of Powers
(x^a)^b = x^(a×b)
Nested exponents: multiply
Special Cases
Zero Exponent:
x^0 = 1 (if x ≠ 0)
Negative Exponent:
x^(-n) = 1/x^n
Fractional Exponent:
x^(1/n) = ⁿ√x
One as Base:
1^n = 1
Quick Examples
Same Base:
2^3 × 2^4 = 2^7 = 128
Same Exponent:
3^2 × 4^2 = (3×4)^2 = 144
Different Base & Exponent:
2^3 × 3^2 = 8 × 9 = 72
Negative Exponents:
2^(-2) × 3^1 = 0.25 × 3 = 0.75
Understanding Exponent Multiplication
What are Exponents?
An exponent tells us how many times to multiply a base number by itself. For example, 2⁴ = 2 × 2 × 2 × 2 = 16. The base is 2, and the exponent is 4.
Key Multiplication Rules
- •Same base: Add the exponents (x^a × x^b = x^(a+b))
- •Same exponent: Multiply the bases ((xy)^a = x^a × y^a)
- •Different bases and exponents: Calculate each term separately
Working with Negative Exponents
Rule: x^(-n) = 1/x^n
A negative exponent means "take the reciprocal and use the positive exponent."
Prime Factorization Method
For complex problems with different bases, you can use prime factorization:
- Find the prime factorization of each base
- Apply the power rules to the prime factors
- Combine like terms using the product rule
- Simplify to get the final answer
Real-World Applications
Computer Science
Powers of 2 are fundamental in computing for memory sizes, data structures, and algorithms.
Finance
Compound interest calculations use exponents to model exponential growth over time.
Science
Scientific notation and exponential functions are used in physics, chemistry, and biology.