What is the Power of a Power Exponent Rule?

The power of a power exponent rule states that when raising a power to another power, you multiply the exponents while keeping the base the same.

In mathematical terms, this rule is expressed as:

*(a^m)ⁿ = a^mn**

Where:

• a is the base.
• m is the inner exponent.
• n is the outer exponent.

Explanation and Examples:

This rule simplifies expressions where an exponent is raised to another exponent. Instead of repeatedly multiplying the base, you can simply multiply the exponents.

Here are some examples:

• *(2³)² = 2³2 = 2⁶ = 64**

  • In this case, 2³ (which is 8) is raised to the power of 2. The rule allows us to directly calculate 2⁶, which is 64.

• *(x⁴)⁵ = x⁴5 = x²⁰**

  • Here, a variable x raised to the power of 4 is then raised to the power of 5. This simplifies to x raised to the power of 20.

• *(5^-2)³ = 5^-23 = 5^-6 = 1/5⁶ = 1/15625**

  • This example includes a negative exponent. The rule still applies: -2 multiplied by 3 equals -6. Remember that a negative exponent means taking the reciprocal (1 divided by the base raised to the positive version of the exponent).

• *(y^1/2)⁴ = y^(1/2)4 = y²**

  • This example includes a fractional exponent. The multiplication works the same way.

Key Takeaways:

• The base remains unchanged.
• Multiply the exponents when raising a power to a power.
• The rule applies to all types of exponents (positive, negative, fractions, etc.).

This rule is a fundamental concept in algebra and is essential for simplifying expressions and solving equations involving exponents.