How Do You Find the Exponent of an Exponent?

To find the exponent of an exponent, you multiply the exponents together. This is often referred to as the "power of a power" rule.

Let's break this down:

The rule can be written as:

*(x^a)^b = x^ab**

Where:

• x is the base.
• a is the inner exponent.
• b is the outer exponent.

Explanation:

Raising a power to another power means you are essentially repeating the base raised to the inner exponent b times. For example:

(x^a)^b = x^a x^a ... * x^a (b times)

When multiplying exponential terms with the same base, you add the exponents. So:

x^a x^a ... x^a (b times) = x^a+a+...+a (b times) = x^ab

Examples:

1. *(2³)² = 2³2 = 2⁶ = 64**
   • Here, we raised 2³ to the power of 2. We multiplied the exponents 3 and 2 to get 6.
2. *(5²)⁴ = 5²4 = 5⁸ = 390625**
   • We raised 5² to the power of 4. We multiplied the exponents 2 and 4 to get 8.
3. *(x⁴)⁵ = x⁴5 = x²⁰**
   • An algebraic example showing the multiplication of exponents.

Key Takeaway:

When you have an expression in the form of (x^a)^b, simply multiply the exponents a and b to get the new exponent. The base, x, remains the same.