1. What is the Exponent Rule with Variables?

The exponent rule with variables demonstrates that (x·a)^b equals x^(a·b). This is an important algebraic identity that shows how exponents interact with multiplication inside parentheses.

2. How Does the Calculator Work?

The calculator uses the exponent rule:

Where:

• \( x \) — Variable value
• \( a \) — Coefficient multiplying the variable
• \( b \) — Exponent applied to the entire expression

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

3. Importance of Exponent Rules

Details: Understanding exponent rules is crucial for simplifying algebraic expressions, solving equations, and working with exponential growth/decay models in various scientific fields.

4. Using the Calculator

Tips: Enter any numerical values for x, a, and b. The calculator will compute both forms of the expression to show they are equivalent. Works with positive and negative numbers (except when x=0 with negative exponents).

5. Frequently Asked Questions (FAQ)

Q1: Does this rule work with negative numbers?
A: Yes, but be cautious with negative bases and fractional exponents, which may produce complex numbers.

Q2: What happens if x=0?
A: When x=0 and b is positive, both sides equal 0. If b is negative, the result is undefined (division by zero).

Q3: Can this rule be extended to more variables?
A: Yes, similar rules apply for expressions like (x·y·a)^b = x^b · y^b · a^b.

Q4: How is this different from (x + a)^b?
A: The rule only applies to multiplication inside the parentheses. For addition, you would need to use the binomial theorem.

Q5: What are practical applications of this rule?
A: This is used in physics equations, compound interest calculations, and anywhere exponential relationships appear in multiplied variables.