1. What is Division of Powers with Variable Bases?

The division of powers with variable bases calculates the result of dividing two exponential terms with different bases and exponents. This operation is fundamental in algebra and appears in various scientific and engineering calculations.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( x \) — First base (variable)
• \( a \) — First exponent (power)
• \( y \) — Second base (variable)
• \( b \) — Second exponent (power)

Explanation: The calculator computes each power separately and then divides the results.

3. Mathematical Explanation

Details: When dividing powers with different bases, we must evaluate each term separately before performing the division. Unlike powers with the same base, we cannot combine the exponents directly.

4. Using the Calculator

Tips: Enter numerical values for all four parameters (x, a, y, b). The denominator (y) cannot be zero when the exponent (b) is negative.

5. Frequently Asked Questions (FAQ)

Q1: Can this be simplified if bases are equal?
A: Yes, if x = y, the expression simplifies to \( x^{a-b} \) by subtracting exponents.

Q2: What if both exponents are equal?
A: If a = b, the expression becomes \( (x/y)^a \), which is the ratio of bases raised to the common exponent.

Q3: How does zero affect the calculation?
A: If x=0 and a>0, numerator is 0. If y=0 and b>0, result is undefined (division by zero).

Q4: What about negative exponents?
A: Negative exponents represent reciprocals (e.g., \( x^{-a} = 1/x^a \)). The calculator handles them correctly.

Q5: Are fractional exponents supported?
A: Yes, the calculator supports fractional exponents, which represent roots (e.g., \( x^{0.5} = \sqrt{x} \)).