Simplifying Exponential Expressions

The question asks us to evaluate the expression $55^{-6} \div 55^5 \times 55^{-8}$. To solve this, we need to apply the fundamental laws of exponents.

Applying Laws of Exponents

We will use the following exponent rules:

• Division Rule: For any non-zero number $a$ and integers $m$ and $n$, $a^m \div a^n = a^{m-n}$. This means when dividing powers with the same base, we subtract the exponent of the divisor from the exponent of the dividend.
• Multiplication Rule: For any non-zero number $a$ and integers $m$ and $n$, $a^m \times a^n = a^{m+n}$. This means when multiplying powers with the same base, we add their exponents.

Step 1: Evaluate the Division

First, consider the division part of the expression: $55^{-6} \div 55^5$. Applying the division rule:

$ 55^{-6} \div 55^5 = 55^{-6 - 5} $

Now, calculate the new exponent:

$ -6 - 5 = -11 $

So, the division simplifies to $55^{-11}$.

Step 2: Evaluate the Multiplication

Next, we take the result from Step 1 ($55^{-11}$) and multiply it by the remaining term, $55^{-8}$. So we need to calculate $55^{-11} \times 55^{-8}$. Applying the multiplication rule:

$ 55^{-11} \times 55^{-8} = 55^{-11 + (-8)} $

Now, calculate the final exponent:

$ -11 + (-8) = -11 - 8 = -19 $

Thus, the expression $55^{-6} \div 55^5 \times 55^{-8}$ simplifies to $55^{-19}$.

Final Answer

The value of the expression $55^{-6} \div 55^5 \times 55^{-8}$ is $55^{-19}$.