Power of a Power Calculator

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Power of a Power Calculator

Result:

function calculatePowerOfPower() { var baseA = parseFloat(document.getElementById("baseA").value); var exponentB = parseFloat(document.getElementById("exponentB").value); var exponentC = parseFloat(document.getElementById("exponentC").value); var resultDiv = document.getElementById("result"); if (isNaN(baseA) || isNaN(exponentB) || isNaN(exponentC)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } // The rule for power of a power is (a^b)^c = a^(b*c) var finalExponent = exponentB * exponentC; var finalResult = Math.pow(baseA, finalExponent); resultDiv.innerHTML = "The calculation is: (" + baseA + "" + exponentB + ")" + exponentC + " = " + baseA + "(" + exponentB + " × " + exponentC + ") = " + baseA + "" + finalExponent + " = " + finalResult + ""; }

Understanding the Power of a Power Rule

In mathematics, an exponent indicates how many times a base number is multiplied by itself. For example, 23 means 2 × 2 × 2, which equals 8.

What is "Power of a Power"?

The "power of a power" rule applies when you have an exponential expression raised to another exponent. It looks like this: (ab)c. Here, 'a' is the base number, 'b' is the first exponent, and 'c' is the second exponent.

The Rule: Multiply the Exponents

The fundamental rule for simplifying a power of a power is to multiply the exponents while keeping the base the same. Mathematically, this is expressed as:

(ab)c = a(b × c)

This rule simplifies complex expressions into a single base with a single exponent, making calculations much easier.

Why Does This Rule Work?

Let's break it down with an example: (23)2.

  • First, calculate the inner power: 23 = 2 × 2 × 2 = 8.
  • Now, raise this result to the outer power: 82 = 8 × 8 = 64.

Using the rule, we multiply the exponents: 2(3 × 2) = 26. And 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64. Both methods yield the same result, demonstrating the validity of the rule.

Practical Examples

  1. Example 1: Simplify (52)3
    • Using the rule: 5(2 × 3) = 56
    • Calculation: 5 × 5 × 5 × 5 × 5 × 5 = 15,625
  2. Example 2: Simplify (104)2
    • Using the rule: 10(4 × 2) = 108
    • Calculation: 100,000,000 (one hundred million)
  3. Example 3: Simplify (3-2)3
    • Using the rule: 3(-2 × 3) = 3-6
    • Calculation: 1 / (36) = 1 / (3 × 3 × 3 × 3 × 3 × 3) = 1 / 729

How to Use the Calculator

Our Power of a Power Calculator makes these calculations straightforward:

  1. Base Number (a): Enter the base number of your expression.
  2. First Exponent (b): Input the first exponent.
  3. Second Exponent (c): Enter the second exponent.
  4. Click "Calculate" to instantly see the simplified expression and its final value.

This tool is ideal for students, educators, or anyone needing to quickly solve or verify power of a power expressions.

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