Exponent Simplification: $69^{-4} \div 69^{10} \times 69^{-17}$

To find the value of the given expression, we need to use the rules of exponents. The expression involves multiplication and division of powers with the same base, which is $69$.

Understanding Exponent Rules

We will use the following rules for exponents:

• Rule 1: Division of Powers with Same Base When dividing powers with the same base, subtract the exponents: $a^m \div a^n = a^{m-n}$
• Rule 2: Multiplication of Powers with Same Base When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$

Applying the Rules Step-by-Step

The given expression is: $69^{-4} \div 69^{10} \times 69^{-17}$

We can solve this by applying the rules sequentially or by combining them. Let's combine the rules. The expression can be rewritten by applying the division rule first, then the multiplication rule. A combined rule for this type of expression is: $a^m \div a^n \times a^p = a^{m-n+p}$

In our expression, the base $a = 69$, and the exponents are $m = -4$, $n = 10$, and $p = -17$.

Now, substitute these values into the combined rule: $69^{-4} \div 69^{10} \times 69^{-17} = 69^{(-4) - 10 + (-17)}$

Calculating the Final Exponent

Let's calculate the exponent value:

Exponent $= -4 - 10 + (-17)$

Exponent $= -4 - 10 - 17$

Exponent $= -14 - 17$

Exponent $= -31$

Final Result

Substituting the calculated exponent back into the expression, we get: $69^{-31}$

Therefore, the value of $69^{-4} \div 69^{10} \times 69^{-17}$ is $69^{-31}$.