Solving $66^{-5} \div 66^{11} \times 66^{-12}$ Using Exponent Rules

This question requires simplifying an expression involving powers with the same base, specifically base 66. We need to use the laws of exponents to find the final value.

Exponent Rules Overview

To solve this, we recall two fundamental rules for exponents:

• Division Rule: When dividing powers with the same base, subtract the exponents. Mathematically, this is represented as: \( a^m \div a^n = a^{m-n} \).
• Multiplication Rule: When multiplying powers with the same base, add the exponents. Mathematically, this is represented as: \( a^m \times a^n = a^{m+n} \).

Step-by-Step Calculation

Let's apply these rules step-by-step to the given expression: \(66^{-5} \div 66^{11} \times 66^{-12}\).

Step 1: Apply the Division Rule

First, we handle the division part of the expression: \(66^{-5} \div 66^{11}\). Using the division rule, we subtract the exponents:

$ 66^{-5} \div 66^{11} = 66^{-5 - 11} = 66^{-16} $

Step 2: Apply the Multiplication Rule

Now, we take the result from Step 1, which is \(66^{-16}\), and multiply it by the next term in the expression, \(66^{-12}\). Using the multiplication rule, we add the exponents:

$ 66^{-16} \times 66^{-12} = 66^{-16 + (-12)} $

Simplifying the exponent:

$ 66^{-16 + (-12)} = 66^{-16 - 12} = 66^{-28} $

Final Value

Therefore, the value of the expression \(66^{-5} \div 66^{11} \times 66^{-12}\) is \(66^{-28}\).