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(4^4)^2 X (4^7)^-2

2 min read Jun 16, 2024
(4^4)^2 X (4^7)^-2

Simplifying Exponential Expressions: (4^4)^2 x (4^7)^-2

This article will guide you through the process of simplifying the expression (4^4)^2 x (4^7)^-2. We'll use the fundamental properties of exponents to reach the simplest form.

Understanding the Rules

Before we begin, let's review the key exponent rules we'll be using:

  • Power of a Power: (a^m)^n = a^(m*n)
  • Negative Exponent: a^-n = 1/a^n

Simplifying the Expression

  1. Apply the Power of a Power Rule:

    • (4^4)^2 = 4^(4*2) = 4^8
    • (4^7)^-2 = 4^(7*-2) = 4^-14

  2. Apply the Negative Exponent Rule:

    • 4^-14 = 1/4^14

  3. Combine the terms:

    • (4^4)^2 x (4^7)^-2 = 4^8 x (1/4^14)

  4. Simplify using the division property of exponents:

    • 4^8 x (1/4^14) = 4^(8-14) = 4^-6

  5. Rewrite with a positive exponent:

    • 4^-6 = 1/4^6

Final Answer

Therefore, the simplified form of (4^4)^2 x (4^7)^-2 is 1/4^6.

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