Exponent Calculator
Solve algebraic powers with step-by-step logs applying negative exponent reciprocals, zero powers, and radical fractional roots.
Computed Power Output
8
Applied Exponent Law
Step-by-Step Power Resolution
Understanding Exponents & Powers
In algebra, an exponent refers to the number of times a number is multiplied by itself. For example, $2^3$ indicates that $2$ is multiplied by itself three times ($2 \times 2 \times 2 = 8$). The value $2$ is the **base**, and $3$ is the **exponent** or **power**.
Core Laws of Exponents
Our calculator walks you through the step-by-step algebraic laws applied to resolve complex powers:
- Zero Power Law: Any non-zero base raised to the power of 0 equals 1 ($b^0 = 1$).
- Negative Exponent Law: A negative exponent represents a division reciprocal. The term $b^{-n}$ translates to $1 / b^n$. For example, $3^-2 = 1 / 3^2 = 1 / 9$.
- Fractional Exponent Law (Radicals): Fractional powers indicate roots. A base raised to $p/q$ corresponds to the $q$-th root of the base raised to the $p$-th power: $b^{p/q} = \sqrt[q]{b^p}$.
Complex and Imaginary Numbers
When you try to evaluate an even root (like a square root $q=2$) of a negative base, the result cannot be expressed as a real number (since no real number multiplied by itself is negative). This calculator uses standard Euler-based complex trigonometry to resolve even fractional roots of negative bases into the complex form $a + bi$.
Frequently Asked Questions
How do you evaluate negative exponents?
To solve a negative exponent, convert it into a fraction by placing 1 in the numerator and the base raised to the positive version of that exponent in the denominator.
What does a fractional exponent mean?
A fractional exponent like x^(p/q) corresponds to taking the q-th root of the base x, and then raising the result to the power of p.
What happens when you raise a negative base to an even fractional root?
In real-number mathematics, this yields an imaginary/complex number because we are evaluating an even root of a negative value. Our calculator detects this case and displays the exact complex result format.
Does this support negative and fractional exponents?
Yes. Handle all exponent types including fractional, negative, and decimal with steps.