Exponent Rules for Division and Multiplication

To find the value of the expression $57^{-9} \div 57^4 \times 57^{-10}$, we apply the basic rules of exponents. Remember these key properties:

• Division of Powers: When dividing two powers with the same base, you subtract the exponents. The formula is: $ a^m \div a^n = a^{m-n} $
• Multiplication of Powers: When multiplying two powers with the same base, you add the exponents. The formula is: $ a^m \times a^n = a^{m+n} $

In our problem, the number 57 is the base for all the powers.

Calculating the Value Step-by-Step

Let's solve the expression $57^{-9} \div 57^4 \times 57^{-10}$ piece by piece:

1. First, perform the division: We have $57^{-9} \div 57^4$. Using the division rule ($a^m \div a^n = a^{m-n}$): $ 57^{-9} \div 57^4 = 57^{(-9) - 4} $ $ = 57^{-13} $
2. Next, perform the multiplication: Now we take the result from the first step, $57^{-13}$, and multiply it by $57^{-10}$. Using the multiplication rule ($a^m \times a^n = a^{m+n}$): $ 57^{-13} \times 57^{-10} = 57^{(-13) + (-10)} $ $ = 57^{-13 - 10} $ $ = 57^{-23} $

So, the final calculated value of the expression is $57^{-23}$.