Simplify the Expression: a^9 Divided by a^4 Using Power Rules

To solve this problem, we'll follow a systematic approach to simplify $\frac{a^9}{a^4}$ using exponent rules:

• Step 1: Identify the expression.
  We are given $\frac{a^9}{a^4}$ and asked to simplify it.
• Step 2: Apply the Quotient Rule for Exponents.
  According to the quotient rule, when dividing two expressions with the same base, we subtract the exponents. Given
  $\frac{a^m}{a^n} = a^{m-n}$,
  we apply this rule to our expression:

$a^{9-4}$

• Step 3: Simplify the Expression.
  We simplify the exponent by performing the subtraction:
  $a^{9-4} = a^5$.

Thus, the simplified form of the expression $\frac{a^9}{a^4}$ is $\mathbf{a^5}$.

Now, let's match our simplified expression with the provided choices:

• Choice 1: $a^{9-4}$.
  This reflects the correct expression before computation.
• Choice 2: $a^{9+4}$.
  Incorrect, as it adds the exponents instead of subtracting.
• Choice 3: $a^{9\times4}$.
  Incorrect, as it multiplies the exponents instead of subtracting.
• Choice 4: $a^{\frac{9}{4}}$.
  Incorrect, as it divides the exponents instead of subtracting.

Hence, the correct choice is Choice 1: $a^{9-4}$, which is the correct representation before simplification.

I am confident in the correctness of this solution.