1. What Are Fractions With Negative Exponents?

A fraction with a negative exponent can be transformed by taking the reciprocal of the fraction and changing the sign of the exponent. This rule is fundamental in algebra and simplifies many mathematical operations.

2. How Does the Calculator Work?

The calculator demonstrates the negative exponent rule:

Where:

• \( a \) — Numerator (unitless)
• \( b \) — Denominator (unitless, cannot be zero)
• \( n \) — Exponent (unitless)

Explanation: The calculator shows both forms of the expression and proves they yield identical results, demonstrating the mathematical equivalence.

3. Importance of Negative Exponents

Details: Understanding negative exponents is crucial for working with scientific notation, simplifying algebraic expressions, and solving equations in physics, chemistry, and engineering.

4. Using the Calculator

Tips: Enter the numerator (a), denominator (b ≠ 0), and exponent (n). The calculator will show both forms of the expression and their identical results.

5. Frequently Asked Questions (FAQ)

Q1: Why does a negative exponent create a reciprocal?
A: This maintains consistency in exponent rules. For example, x² ÷ x³ = x⁻¹ = 1/x, showing negative exponents naturally represent reciprocals.

Q2: What if both the fraction and exponent are negative?
A: Two negatives cancel out: (-a/b)⁻ⁿ = (-b/a)ⁿ. The sign depends on whether n is odd or even.

Q3: Can the denominator be zero?
A: No, division by zero is undefined in mathematics. The calculator requires a non-zero denominator.

Q4: How does this apply to unit fractions?
A: For 1/b⁻ⁿ, it becomes bⁿ. The rule works the same way with numerator 1.

Q5: What about fractional exponents?
A: The rule applies to all real exponents: (a/b)⁻ˣ = (b/a)ˣ, whether x is integer, fractional, or decimal.