Power & Exponent Calculator

The calculator operates using two distinct computational engines depending on your selected mode: Standard Power and Modular Exponentiation.

1. Standard Power ($b^x$)

Exponentiation is a mathematical operation involving a base ($b$) and an exponent ($x$). The fundamental rule changes depending on the nature of the exponent:

• Positive Integers: Exponentiation corresponds to repeated multiplication.
• $b^x = b \times b \times \dots \times b$ ($x$ times)
• Negative Exponents: A negative exponent is defined as the reciprocal of the base raised to the absolute value of the exponent.
• $b^{-x} = \frac{1}{b^x}$
• Fractional (Rational) Exponents: Fractional exponents represent roots. The calculator uses a continuous fraction algorithm to convert decimals like 0.5 into fractions like 1/2.
• $b^{m/n} = \sqrt[n]{b^m}$
• The Zero Rule: Any non-zero base raised to the power of $0$ is strictly $1$. ($b^0 = 1$).

For massive integer inputs (e.g., $3^{150}$), the engine automatically drops down into JavaScript's BigInt processing layer to calculate the exact digits without truncating them into standard floating-point approximation limits.

2. Modular Exponentiation ($b^e \pmod m$)

Calculating $b^e \pmod m$ directly by first finding $b^e$ and then finding the remainder is practically impossible for large cryptographic values, as $b^e$ becomes unimaginably huge and causes memory overflow.

Instead, this tool uses the Right-to-Left Square-and-Multiply Algorithm, keeping the numbers strictly bounded by the modulus $m$ at every single step:

1. Initialize a result = 1, and set the initial base value current_base = b % m.
2. Convert the exponent $e$ into its binary representation.
3. Loop through the binary bits of the exponent:

• If the current bit is 1 (odd), multiply the result by the current_base, then modulo $m$.
• Square the current_base and modulo $m$.
• Shift the exponent right (divide by 2).

4. The loop terminates when the exponent reaches $0$, returning the final congruence solution instantly, even for massive 2048-bit numbers.