Exponent Calculator
Calculate exponents (powers) and roots instantly. Supports negative, fractional, and decimal exponents.
Formula
x^y
Exponentiation is repeated multiplication. b^n means multiplying b by itself n times.
Worked Examples
Example 1: Positive Integer Exponent
Problem:Evaluate 2^3, then 5^4.
Solution:A positive whole exponent means repeated multiplication of the base. 2^3 = 2 ร 2 ร 2 = 8 5^4 = 5 ร 5 ร 5 ร 5 = 625 Each extra power multiplies by the base once more, which is why growth is so fast: 5^5 would already be 3,125.
Result:2^3 = 8 and 5^4 = 625
Example 2: Negative Exponent
Problem:Evaluate 2^-2 and 10^-3.
Solution:A negative exponent means the reciprocal of the positive power: b^-n = 1 / b^n. 2^-2 = 1 / 2^2 = 1/4 = 0.25 10^-3 = 1 / 10^3 = 1/1000 = 0.001 The base is never negated - only the exponent moves to the denominator, so 2^-2 is 0.25, not -4.
Result:2^-2 = 0.25 and 10^-3 = 0.001
Example 3: Fractional Exponent as a Root
Problem:Evaluate 9^(1/2), 27^(1/3) and 16^(3/4).
Solution:A fractional exponent m/n means the nth root raised to the mth power. 9^(1/2) = the square root of 9 = 3 27^(1/3) = the cube root of 27 = 3 16^(3/4) = (the fourth root of 16)^3 = 2^3 = 8 Check: 8^(4/3) = (cube root of 8)^4 = 2^4 = 16, which reverses the last one.
Result:9^(1/2) = 3, 27^(1/3) = 3, 16^(3/4) = 8
Frequently Asked Questions
What is a number raised to the power of 0?
Any non-zero number raised to the power of 0 is 1. (0^0 is indeterminate).
What does a negative exponent mean?
A negative exponent indicates the reciprocal. x^-n equals 1 divided by x^n.
What is a fractional exponent?
Fractional exponents represent roots. x^(1/n) is the nth root of x. x^(1/2) is the square root.
What is the Power Rule?
When raising a power to another power, you multiply the exponents: (x^a)^b = x^(a*b).
What is the Product Rule?
When multiplying terms with the same base, you add the exponents: x^a * x^b = x^(a+b).
How do exponents relate to logarithms?
Logarithms are the inverse of exponentiation. If b^x = y, then log_b(y) = x.
Real-world examples of exponential growth?
Compound interest, population growth, viral spread, and computing processing power (Moore's Law).
Background & Theory
Exponentiation:
Exponentiation is a mathematical operation involving two numbers, the base (b) and the exponent (n). It is written as bโฟ.
- bโฟ: "b raised to the power of n"
- Definition: Multiply 'b' by itself 'n' times.
Rules:
- Product Rule: aแต ร aโฟ = aแตโบโฟ
- Quotient Rule: aแต / aโฟ = aแตโปโฟ
- Power Rule: (aแต)โฟ = aแตโฟ
- Negative Exponent: aโปโฟ = 1/aโฟ
- Zero Exponent: aโฐ = 1 (for a โ 0)
Roots:
The nth root of a number x is a number r which, when raised to the power n, yields x.
- โฟโx = r โบ rโฟ = x