Exponent Solver Calculator
Solve exponential equations, simplify powers, and compute roots with step-by-step solutions
Tip: Use ^ for exponents (e.g., 2^3 = 8) or sqrt() for roots (e.g., sqrt(9) = 3).
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Understanding Exponents
What Are Exponents?
An exponent indicates how many times a number (the base) is multiplied by itself. For example, 2³ means 2 × 2 × 2 = 8. Exponents follow specific rules that make calculations with large numbers more manageable.
Exponent Rules
Key exponent rules include: Product Rule (xᵃ × xᵇ = xᵃ⁺ᵇ), Quotient Rule (xᵃ ÷ xᵇ = xᵃ⁻ᵇ), Power Rule ((xᵃ)ᵇ = xᵃᵇ), Zero Exponent (x⁰ = 1), and Negative Exponents (x⁻ᵃ = 1/xᵃ).
Real-World Applications
Exponents are used in various fields including:
- Compound interest calculations in finance
- Population growth models in biology
- Radioactive decay in physics
- Computer science algorithms and big-O notation
- Scientific notation for very large or small numbers
Exponent Rules and Examples
| Rule Name | Expression | Example |
|---|---|---|
| Product Rule | xᵃ × xᵇ = xᵃ⁺ᵇ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient Rule | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 5⁶ ÷ 5² = 5⁴ = 625 |
| Power Rule | (xᵃ)ᵇ = xᵃᵇ | (3²)³ = 3⁶ = 729 |
| Zero Exponent | x⁰ = 1 | 7⁰ = 1 |
| Negative Exponent | x⁻ᵃ = 1/xᵃ | 2⁻³ = 1/8 = 0.125 |
| Fractional Exponent | x¹/ⁿ = ⁿ√x | 8¹/³ = ³√8 = 2 |
Frequently Asked Questions
What's the difference between an exponent and a power?
In the expression xⁿ, x is called the base and n is called the exponent or power. The term "power" sometimes refers to the entire expression (xⁿ) while "exponent" specifically refers to the n.
How do I handle negative exponents?
A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x⁻ⁿ = 1/xⁿ. This means 2⁻³ = 1/2³ = 1/8 = 0.125.
What about fractional exponents?
Fractional exponents represent roots. The denominator of the fraction indicates the root, while the numerator indicates the power. For example, x^(m/n) = (ⁿ√x)ᵐ.
Can this calculator solve exponential equations?
Yes! Switch to "Exponential Equations" mode to solve equations like 2^x = 32 or e^(3x) = 20. The calculator will provide the solution with steps.
How accurate are the calculations?
The calculator provides precise calculations up to 16 decimal places. You can control the displayed precision using the decimal places option.