Understanding the Exponent Problem

The question asks us to simplify the mathematical expression $24^{-10} \div 24^{16} \times 24^{-7}$. This involves using the rules of exponents, specifically for division and multiplication of powers with the same base.

Exponent Rules Review

To solve this, we need to remember two fundamental rules of exponents:

• Division Rule: When dividing powers with the same base, you 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, you add the exponents. Mathematically, this is represented as: $ a^m \times a^n = a^{m+n} $

Step-by-Step Simplification

Let's apply these rules to the given expression $24^{-10} \div 24^{16} \times 24^{-7}$. We'll work from left to right.

1. Step 1: Apply the Division Rule

   First, we handle the division part: $24^{-10} \div 24^{16}$. Using the division rule ($a^m \div a^n = a^{m-n}$), we subtract the exponents:

   $ 24^{-10} \div 24^{16} = 24^{-10 - 16} = 24^{-26} $
2. Step 2: Apply the Multiplication Rule

   Now, we take the result from Step 1 and multiply it by the next term: $24^{-26} \times 24^{-7}$. Using the multiplication rule ($a^m \times a^n = a^{m+n}$), we add the exponents:

   $ 24^{-26} \times 24^{-7} = 24^{-26 + (-7)} = 24^{-26 - 7} = 24^{-33} $

Final Result

After applying the rules of exponents step-by-step, the simplified value of the expression $24^{-10} \div 24^{16} \times 24^{-7}$ is $24^{-33}$.