Exponent Calculator

Calculate powers, roots, negative, and fractional exponents with ease.

Calculator Inputs

Base (a)

Exponent (n)

Decimal Places 0 1 2 3 4 5 6 7 8 9 10

Options

Show step-by-step Use rational mode

Calculation Result

What Is an Exponent?

An exponent represents the number of times a base number is multiplied by itself. In the expression an, 'a' is the base, and 'n' is the exponent or power. For example, 34 means multiplying 3 by itself four times: 3 × 3 × 3 × 3 = 81.

Rules of Exponents

Exponents follow several fundamental rules that simplify mathematical expressions:

• Product Rule: When multiplying two powers with the same base, you add the exponents: am × an = am+n.
• Quotient Rule: When dividing two powers with the same base, you subtract the exponents: am / an = am-n.
• Power of a Power Rule: To raise a power to another power, you multiply the exponents: (am)n = am×n.
• Power of a Product Rule: To raise a product to a power, you raise each factor to that power: (a × b)n = an × bn.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is 1: a0 = 1 (where a ≠ 0).
• Negative Exponent Rule: A negative exponent indicates a reciprocal: a-n = 1 / an.
• Fractional Exponent Rule: An exponent of the form p/q represents a root: ap/q = q√(ap).

How Fractional and Negative Exponents Work

While positive integer exponents are straightforward, fractional and negative exponents can seem complex at first glance. Our calculator handles them seamlessly, but here’s a breakdown:

Negative Exponents: A negative exponent means "take the reciprocal of the base raised to the positive exponent." For example, to calculate 10-2, you first calculate 102 (which is 100) and then take its reciprocal, resulting in 1/100 or 0.01.

Fractional Exponents: A fractional exponent like p/q combines both a power and a root. The denominator (q) indicates the root, and the numerator (p) indicates the power. For instance, to calculate 272/3:

1. Take the cube root (denominator is 3) of the base: 3√27 = 3.
2. Raise the result to the power of the numerator (2): 32 = 9.

This calculator can simplify these calculations and show you the steps involved.

Common Mistakes When Using Exponents

• Negative Bases: Be careful with negative bases. (-3)2 is (-3) × (-3) = 9, but -32 is -(3 × 3) = -9. The parentheses are critical.
• Fractional Exponents on Negative Bases: You cannot take an even root (like a square root) of a negative number in the real number system. Therefore, (-4)1/2 has no real result.
• The 0⁰ Case: The expression 00 is an indeterminate form in mathematics. It doesn't have a single defined value, so this calculator flags it as undefined.

Disclaimer: This calculator provides mathematical results and educational explanations. Verify results for critical applications.