Simplify the Power Expression: 5^6 ÷ 5^13

To solve this problem, we'll use the quotient rule for exponents:

The formula to use is $\frac{a^m}{a^n} = a^{m-n}$, applicable when the bases are the same.

Let's apply this step-by-step:

• Step 1: We have the expression $\frac{5^6}{5^{13}}$. Identify $m = 6$ and $n = 13$ with base $5$.
• Step 2: Apply the formula $\frac{5^6}{5^{13}} = 5^{6-13}$.
• Step 3: Simplify the exponent: $6 - 13 = -7$, so $5^{6-13} = 5^{-7}$.
• Step 4: Recognize that a negative exponent means a reciprocal: $5^{-7} = \frac{1}{5^7}$.

Therefore, the solution to the problem is $\frac{1}{5^7}$.

Now, considering the answer choices:

• Choice 1: $5^7$, not correct as it doesn't reflect the negative exponent.
• Choice 2: $\frac{1}{5^7}$, correct since it matches our simplified solution.
• Choice 3: $5^{19}$, incorrect; this would imply adding exponents.
• Choice 4: a'+b' are correct. This is not relevant to our examined choices.

Thus, the correct option is Choice 2: $\frac{1}{5^7}$.