Simplify the Expression: 12⁵ ÷ 12⁸ Using Laws of Exponents

To simplify the expression $\frac{12^5}{12^8}$, we'll follow these steps:

• Step 1: Apply the quotient rule for exponents.
• Step 2: Simplify and interpret the result using negative exponents if necessary.

Let's work through each step:

Step 1: Apply the quotient rule for exponents.
We are given the expression $\frac{12^5}{12^8}$. According to the quotient rule for exponents, $\frac{a^m}{a^n} = a^{m-n}$, so we have:

$\frac{12^5}{12^8} = 12^{5-8} = 12^{-3}$

Step 2: Simplify and interpret.
The result $12^{-3}$ can be expressed using the concept of negative exponents $a^{-n} = \frac{1}{a^n}$:

$12^{-3} = \frac{1}{12^3}$

Therefore, both expressions $12^{-3}$ and $\frac{1}{12^3}$ are equivalent.

Matching with the provided choices:
- Choice 1: $12^{-3}$ - This matches our first result.
- Choice 2: $\frac{1}{12^3}$ - This matches our interpretation of the negative exponent.

Choice 4 states: "a'+b' are correct," which refers to both expressions being correct representations. Therefore, the correct answer is "a'+b' are correct."