Simplify the Expression: 7^10 Divided by 7^13 Using Exponent Rules

To solve this problem, we'll follow these steps:

• Step 1: Apply the quotient rule for exponents
• Step 2: Simplify the resulting expression
• Step 3: Compare the simplified form with the given choices

Now, let's work through each step:
Step 1: Given the expression $\frac{7^{10}}{7^{13}}$, use the quotient rule for exponents, which states $\frac{a^m}{a^n} = a^{m-n}$.
Step 2: Apply this rule to get $7^{10-13} = 7^{-3}$.
Step 3: Rewrite $7^{-3}$ using the rule for negative exponents, which is $a^{-n} = \frac{1}{a^n}$. Therefore, $7^{-3} = \frac{1}{7^3}$.

Comparing with the provided answer choices, the correct choice is:

• Choice 2: $\frac{1}{7^3}$

Therefore, the solution to the problem is $\frac{1}{7^3}$, confirming the correctness of the derived expression and matching the provided answer.