(9^5 - 9^4)/8 =

1. Translate the problem requirements: We need to simplify the expression \(\frac{\mathrm{9}^5 - \mathrm{9}^4}{8}\) and match it to one of the given answer choices
2. Factor out the common term: Identify that both terms in the numerator share \(\mathrm{9}^4\) as a common factor
3. Simplify the factored expression: Reduce the expression after factoring to get the final answer
4. Verify against answer choices: Confirm our simplified result matches one of the given options

Execution of Strategic Approach

1. Translate the problem requirements

We need to simplify the expression \(\frac{\mathrm{9}^5 - \mathrm{9}^4}{8}\) and match it to one of the given answer choices.

Let's break this down in plain English: We have a fraction where the top part (numerator) contains two powers of 9 being subtracted from each other, and the bottom part (denominator) is 8. Our job is to simplify this expression to see which answer choice it matches.

Process Skill: TRANSLATE - Converting the mathematical expression into a clear understanding of what we need to accomplish

2. Factor out the common term

Now let's look at the numerator: \(\mathrm{9}^5 - \mathrm{9}^4\)

Think of this like having money in your pocket. If you have \(\mathrm{9}^5\) dollars and \(\mathrm{9}^4\) dollars, what's the difference? Well, \(\mathrm{9}^5\) means \(9\times9\times9\times9\times9\), and \(\mathrm{9}^4\) means \(9\times9\times9\times9\).

Notice that both terms have \(9\times9\times9\times9\) in common - that's \(\mathrm{9}^4\)! So we can factor this out:

In everyday terms: \(\mathrm{9}^5\) is the same as \(\mathrm{9}^4 \times 9\). So our subtraction becomes:
\(\mathrm{9}^5 - \mathrm{9}^4 = (\mathrm{9}^4 \times 9) - (\mathrm{9}^4 \times 1) = \mathrm{9}^4 \times (9 - 1) = \mathrm{9}^4 \times 8\)

Mathematically: \(\mathrm{9}^5 - \mathrm{9}^4 = \mathrm{9}^4(9 - 1) = \mathrm{9}^4 \times 8\)

Process Skill: SIMPLIFY - Recognizing and extracting common factors to make the expression easier to work with

3. Simplify the factored expression

Now our original expression becomes much simpler:

Original: \(\frac{\mathrm{9}^5 - \mathrm{9}^4}{8}\)
After factoring: \(\frac{\mathrm{9}^4 \times 8}{8}\)

This is like having \((\mathrm{9}^4 \times 8)\) pieces of something and dividing them into 8 equal groups. Each group would have \(\mathrm{9}^4\) pieces.

When we divide something by the same number we multiplied it by, they cancel out:
\(\frac{\mathrm{9}^4 \times 8}{8} = \mathrm{9}^4\)

The 8 in the numerator and the 8 in the denominator cancel each other out, leaving us with just \(\mathrm{9}^4\).

4. Verify against answer choices

Our simplified result is \(\mathrm{9}^4\).

Looking at the answer choices:

1. \(\mathrm{9}^4\) ✓ - This matches our result exactly
2. \(\frac{\mathrm{9}^4}{8}\) - This is what we would have if we didn't factor correctly
3. \(\mathrm{9}^3\) - This would be if we made an error in our exponent arithmetic
4. \(\frac{9}{8}\) - This is much too small
5. \(\frac{1}{8}\) - This is also much too small

Final Answer

The answer is A. \(\mathrm{9}^4\)

We successfully simplified \(\frac{\mathrm{9}^5 - \mathrm{9}^4}{8}\) by factoring out \(\mathrm{9}^4\) from the numerator, which gave us \(\frac{\mathrm{9}^4 \times 8}{8}\), and then the 8's cancelled out to leave us with \(\mathrm{9}^4\).

Common Faltering Points

Errors while devising the approach

1. Not recognizing the factoring opportunity: Students may see \(\mathrm{9}^5 - \mathrm{9}^4\) and immediately think they need to calculate each power separately (\(\mathrm{9}^5 = 59,049\) and \(\mathrm{9}^4 = 6,561\)), leading to unwieldy arithmetic. They miss that both terms share a common factor of \(\mathrm{9}^4\), which is the key insight for elegant simplification.

2. Attempting to use difference of powers formulas incorrectly: Some students might try to apply formulas like \(\mathrm{a}^n - \mathrm{b}^n\) factorization, not recognizing that this is actually a simpler case where we can factor out the smaller power directly.

Errors while executing the approach

1. Factoring errors with exponents: When factoring \(\mathrm{9}^5 - \mathrm{9}^4\), students might incorrectly write it as \(\mathrm{9}^4(\mathrm{9}^5 - \mathrm{9}^4)\) instead of the correct \(\mathrm{9}^4(9 - 1)\). They fail to properly apply the rule that when factoring out \(\mathrm{9}^4\) from \(\mathrm{9}^5\), what remains is \(\mathrm{9}^{(5-4)} = \mathrm{9}^1 = 9\).

2. Arithmetic mistakes in simplification: Students might correctly factor to get \(\mathrm{9}^4(9 - 1)\) but then make basic arithmetic errors, such as calculating \((9 - 1)\) as something other than 8, or incorrectly canceling terms when they have \(\frac{\mathrm{9}^4 \times 8}{8}\).

3. Cancellation confusion: When students reach \(\frac{\mathrm{9}^4 \times 8}{8}\), they might incorrectly cancel the \(\mathrm{9}^4\) with the 8 in the denominator, or they might think that \(\frac{8}{8}\) equals 0 instead of 1, leading to incorrect final expressions.

Errors while selecting the answer

1. Stopping at an intermediate step: Students might correctly factor the numerator to get \(\mathrm{9}^4 \times 8\) but then select answer choice B (\(\frac{\mathrm{9}^4}{8}\)), forgetting to complete the final cancellation step where the 8 in the numerator cancels with the 8 in the denominator.