1. What is a Fractional Exponent?

A fractional exponent represents both a power and a root. The expression \( base^{p/q} \) is equivalent to taking the q-th root of the base raised to the p-th power.

2. How Does the Calculator Work?

The calculator uses the fractional exponent formula:

Where:

• \( base \) — The base number
• \( p \) — The numerator of the exponent
• \( q \) — The denominator of the exponent (must be non-zero)

Explanation: The formula shows that a fractional exponent can be computed by first raising the base to the numerator power, then taking the denominator root of the result.

3. Importance of Fractional Exponents

Details: Fractional exponents are fundamental in mathematics, physics, and engineering. They allow for compact representation of roots and powers, and are essential in calculus, exponential growth models, and many scientific formulas.

4. Using the Calculator

Tips: Enter the base value, the numerator (p) and denominator (q) of the exponent. The denominator must be non-zero. All values can be positive or negative, except the denominator cannot be zero.

5. Frequently Asked Questions (FAQ)

Q1: What does a negative fractional exponent mean?
A: A negative fractional exponent \( base^{-p/q} \) equals \( 1/(base^{p/q}) \). It represents the reciprocal of the positive fractional exponent.

Q2: Can the base be negative?
A: Yes, but with limitations. For even denominators, negative bases yield complex results. The calculator provides real solutions when possible.

Q3: What's the difference between p/q and q√(base^p)?
A: They are mathematically equivalent. The fractional exponent notation is often more convenient for algebraic manipulation.

Q4: How are fractional exponents used in real life?
A: They appear in compound interest calculations, physics laws (like inverse-square laws), and many growth/decay models.

Q5: What happens when q=1?
A: The expression simplifies to \( base^p \), a simple power function without any root component.