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Laws of Exponents
calculators.exponent.powersOf2
المصادر والمنهجية
xⁿ = x × x × ... × x (n times)Exponential calculation
How to Use the Exponent Calculator
The exponent calculator computes any number raised to any power, including negative exponents, fractional exponents, and zero. See the expanded form of your calculation and understand the laws of exponents with interactive examples.
Understanding Exponents
An exponent indicates how many times to multiply a number (the base) by itself:
2⁵ = 2 × 2 × 2 × 2 × 2 = 32
Here, 2 is the base and 5 is the exponent (or power).
Laws of Exponents
- Product Rule: xᵃ × xᵇ = xᵃ⁺ᵇ
- Quotient Rule: xᵃ ÷ xᵇ = xᵃ⁻ᵇ
- Power Rule: (xᵃ)ᵇ = xᵃˣᵇ
- Zero Exponent: x⁰ = 1 (when x ≠ 0)
- Negative Exponent: x⁻ⁿ = 1/xⁿ
- Fractional Exponent: x^(1/n) = ⁿ√x
Negative Exponents
A negative exponent means "take the reciprocal":
- 2⁻³ = 1/2³ = 1/8 = 0.125
- 10⁻² = 1/100 = 0.01
Fractional Exponents
Fractional exponents represent roots:
- x^(1/2) = √x (square root)
- x^(1/3) = ³√x (cube root)
- x^(2/3) = ³√(x²) or (³√x)²
Large Number Handling
For very large results, the calculator displays results in scientific notation. For example, 10²⁰ = 1 × 10²⁰.
Related Calculators
For logarithms (the inverse of exponents), use our log calculator. For roots, try our square root calculator.
الأسئلة الشائعة
A negative exponent means take the reciprocal. x⁻ⁿ = 1/xⁿ. For example, 2⁻³ = 1/2³ = 1/8. This extends the pattern: 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4.
By the quotient rule: xⁿ / xⁿ = xⁿ⁻ⁿ = x⁰. Since any number divided by itself equals 1, x⁰ = 1. This holds for any non-zero base. Note: 0⁰ is typically defined as 1 in combinatorics but is sometimes left undefined.
A fractional exponent represents a root. x^(1/n) = ⁿ√x (the nth root). For example, 8^(1/3) = ³√8 = 2. Combined: x^(m/n) = (ⁿ√x)^m = ⁿ√(x^m). So 8^(2/3) = (³√8)² = 2² = 4.
Add the exponents: xᵃ × xᵇ = xᵃ⁺ᵇ. For example, 2³ × 2⁴ = 2⁷ = 128. This works because youre multiplying (2×2×2) × (2×2×2×2) = 2×2×2×2×2×2×2 = seven 2s multiplied together.
Subtract the exponents: xᵃ / xᵇ = xᵃ⁻ᵇ. For example, 2⁷ / 2³ = 2⁴ = 16. If the exponent in the denominator is larger, the result is negative: 2³ / 2⁷ = 2⁻⁴ = 1/16.
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