1. What is the Fraction Exponents Rule?

The fraction exponents rule states that a fraction raised to a power is equal to the numerator raised to that power divided by the denominator raised to that power. This is a fundamental rule in algebra that helps simplify complex expressions.

2. How Does the Calculator Work?

The calculator demonstrates the mathematical identity:

Where:

• \( a \) — Numerator (any real number)
• \( b \) — Denominator (any non-zero real number)
• \( n \) — Exponent (any real number)

Explanation: The calculator shows both forms of the expression are mathematically equivalent by computing them separately and showing they yield the same result.

3. Mathematical Proof

Proof: The rule can be derived from the definition of exponents and fraction multiplication: \[ \left(\frac{a}{b}\right)^n = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \cdots \times \frac{a}{b}}_{n \text{ times}} = \frac{a^n}{b^n} \]

4. Using the Calculator

Tips: Enter any numbers for numerator and exponent. Denominator must be non-zero. The calculator will show both forms of the expression and verify they are equal.

5. Frequently Asked Questions (FAQ)

Q1: Does this work for negative exponents?
A: Yes, the rule holds for all real exponents, including negative ones. A negative exponent simply means taking the reciprocal.

Q2: What about fractional exponents?
A: The rule works for fractional exponents as well. For example, (a/b)^(1/2) = √a/√b.

Q3: Why must the denominator be non-zero?
A: Division by zero is undefined in mathematics, so the denominator can never be zero.

Q4: Can this be extended to more complex fractions?
A: Yes, the rule applies to any fraction, no matter how complex the numerator and denominator are.

Q5: How is this useful in real applications?
A: This rule is fundamental in simplifying algebraic expressions, solving equations, and appears frequently in scientific calculations.