1. What is the Power of a Power Rule?

The power of a power rule is a fundamental exponent rule in algebra that states when you raise a power to another power, you multiply the exponents. The general form is \((a^m)^n = a^{m \times n}\).

2. How Does the Calculator Work?

The calculator applies the power of a power rule:

Where:

• \( a \) — The base number
• \( m \) — First exponent
• \( n \) — Second exponent

Explanation: The rule simplifies complex exponential expressions by multiplying the exponents when one power is raised to another.

3. Importance of the Rule

Details: This rule is essential for simplifying exponential expressions, solving equations with exponents, and is foundational for more advanced mathematics including calculus and scientific notation.

4. Using the Calculator

Tips: Enter the base number and both exponents. The calculator will compute the simplified form using the power of a power rule.

5. Frequently Asked Questions (FAQ)

Q1: Does this rule work with negative exponents?
A: Yes, the rule applies regardless of whether the exponents are positive, negative, or fractions.

Q2: What if the base is negative?
A: The rule still applies, but be cautious with even/odd exponents which affect the sign of the result.

Q3: Can this be applied to variables?
A: Yes, the rule works the same way with algebraic expressions (e.g., \((x^2)^3 = x^6\)).

Q4: How is this different from the product of powers rule?
A: The product rule \(a^m \times a^n = a^{m+n}\) adds exponents when multiplying same bases, while power of power multiplies exponents.

Q5: Are there any exceptions to this rule?
A: The rule holds for all real numbers when the base is positive. With negative bases, fractional exponents may produce complex numbers.