Exponent Calculator

Understanding Exponents

Exponentiation is a mathematical operation represented as \(a^n\), where a is the base and n is the exponent. When n is a positive integer, exponentiation corresponds to multiplying the base by itself n times:

\[ a^n = a \times a \times \dots \times a \quad (\text{n times}) \]

Negative bases are allowed but imaginary results are not supported. Fractional exponents are possible if entered in decimal form.

Basic Exponent Rules

1. Multiplying with the same base: Add the exponents.
   \[ a^m \times a^n = a^{m+n} \]
   Example: \(2^2 \times 2^4 = 2^{2+4} = 2^6 = 64\)
2. Negative exponents: Take the reciprocal of the base and use the positive exponent.
   \[ a^{-n} = \frac{1}{a^n} \]
   Example: \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
3. Dividing with the same base: Subtract the exponents.
   \[ \frac{a^m}{a^n} = a^{m-n} \]
   Example: \(\frac{2^2}{2^4} = 2^{2-4} = 2^{-2} = \frac{1}{4}\)
4. Power of a power: Multiply the exponents.
   \[ (a^m)^n = a^{m \cdot n} \]
   Example: \((2^2)^4 = 2^{2 \cdot 4} = 2^8 = 256\)
5. Power distributed over multiplication:
   \[ (a \cdot b)^n = a^n \cdot b^n \]
   Example: \((2 \cdot 4)^2 = 2^2 \cdot 4^2 = 4 \cdot 16 = 64\)
6. Power distributed over division:
   \[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]
   Example: \(\left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25}\)
7. Exponent of 1: The base remains unchanged.
   \[ a^1 = a \]
8. Exponent of 0: Any nonzero base raised to 0 equals 1.
   \[ a^0 = 1 \]
   Explanation: Using the law \(a^n \cdot a^0 = a^{n+0} = a^n\), the only way to remain consistent is \(a^0 = 1\).
9. Fractional exponents: Represent roots of the base.
   \[ a^{1/n} = \sqrt[n]{a}, \quad a^{m/n} = \left(\sqrt[n]{a}\right)^m \]

Exponents with Negative Bases

• Positive integer exponents follow the same magnitude rules; the sign depends on whether the exponent is even or odd.
• Fractional exponents with negative bases involve imaginary numbers, which are not supported by this calculator.