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How do you evaluate a power raised to another power?

Published in Exponent Rules 2 mins read

To evaluate a power raised to another power, you multiply the exponents while keeping the base the same. This is known as the power of a power rule.

Understanding the Power of a Power Rule

The power of a power rule is a fundamental concept in mathematics, particularly when dealing with exponents. The rule essentially provides a shortcut for simplifying expressions where a power is raised to yet another power.

Definition

The rule states that when you have an expression in the form of (am)n, it is simplified to am*n.
In simpler terms:

  • Base: The base remains unchanged.
  • Exponents: The exponents are multiplied together.

Formula

The general formula can be expressed as:

*(am)n = amn**

Where:

  • 'a' represents the base
  • 'm' and 'n' represent the exponents

Step-by-step process

Here’s a breakdown of how to apply this rule:

  1. Identify the Base: Determine the base (the number being raised to a power).
  2. Identify the Exponents: Find the two exponents.
  3. Multiply the Exponents: Multiply the exponents together.
  4. Keep the Base: Retain the original base.
  5. Write the Simplified Expression: Write the base raised to the new exponent, which is the result of the multiplication.

Examples

Here are a few examples illustrating the application of the power of a power rule:

  • *(23)2 = 232 = 26 = 64**
    • Here, the base is 2, and the exponents are 3 and 2. We multiply the exponents (3 * 2) to get 6.
  • *(52)3 = 523 = 56 = 15625**
    • In this example, 5 is the base, and 2 and 3 are the exponents. Multiplying 2 and 3 gives 6.
  • *(x4)5 = x45 = x20**
    • Here, the base is the variable x, and the exponents are 4 and 5. Multiplying 4 and 5 gives 20.

Practical Insights

  • This rule significantly simplifies complex expressions involving nested exponents.
  • It applies to both numerical and algebraic expressions.
  • It reduces the need for repeated multiplication.

Table Summary

Original Expression Apply Power of a Power Rule Simplified Expression
(23)2 23*2 26 = 64
(52)3 52*3 56 = 15625
(x4)5 x4*5 x20

By consistently applying this rule, you can easily handle expressions that involve raising one power to another.

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