1. What Is Powers Of Powers?

The powers of powers rule is a fundamental exponent rule in mathematics that simplifies expressions where a power is raised to another power. This rule states that when you have a power raised to another power, you multiply the exponents.

2. How Does The Calculator Work?

The calculator uses the powers of powers formula:

Where:

• \( a \) — Base number
• \( m \) — First exponent
• \( n \) — Second exponent
• \( m \times n \) — Product of exponents

Explanation: This rule allows us to simplify complex exponential expressions by multiplying the exponents rather than performing multiple power operations.

3. Importance Of Powers Of Powers

Details: The powers of powers rule is essential in algebra, calculus, and scientific calculations. It simplifies complex exponential expressions, makes calculations more efficient, and is fundamental to understanding exponential growth and decay in various fields including physics, engineering, and finance.

4. Using The Calculator

Tips: Enter the base number and both exponents. The calculator will compute the result by multiplying the exponents and raising the base to that product. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: Does this rule work with negative exponents?
A: Yes, the powers of powers rule works with negative exponents. For example, (a⁻²)³ = a⁻⁶.

Q2: What if the base is zero?
A: If the base is zero and the exponents are positive, the result is zero. However, zero raised to a negative exponent is undefined.

Q3: Can this rule be applied to fractional exponents?
A: Yes, the rule works with fractional exponents. For example, (a¹/²)² = a¹ = a.

Q4: Is this rule commutative?
A: The multiplication of exponents is commutative, so (a^m)^n = (a^n)^m = a^(m×n).

Q5: How is this different from product of powers rule?
A: Product of powers rule: a^m × a^n = a^(m+n). Powers of powers rule: (a^m)^n = a^(m×n). They are different exponent rules for different situations.