99,999^2 - 1^2 =

1. Translate the problem requirements: We need to calculate \(99,999^2 - 1^2\) and express the result in a form that matches one of the given answer choices, all of which involve powers of 10.
2. Recognize the algebraic pattern: Identify this as a difference of squares (\(a^2 - b^2\)) which factors as \((a + b)(a - b)\), allowing us to avoid calculating large squares directly.
3. Express numbers using powers of 10: Rewrite 99,999 in terms of powers of 10 to make the factorization cleaner and match the answer choice formats.
4. Apply the difference of squares formula: Factor the expression and simplify to match one of the provided answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking for. We need to calculate \(99,999^2 - 1^2\) and find which answer choice matches our result.

Looking at the answer choices, they all involve powers of 10 like \(10^4\), \(10^5\), etc. This tells us we should express our final answer using powers of 10 rather than calculating the actual numerical value.

Since \(1^2 = 1\), our expression becomes: \(99,999^2 - 1\)

Process Skill: TRANSLATE - Converting the calculation requirement into a form that matches the answer choice format

2. Recognize the algebraic pattern

Here's the key insight: instead of trying to calculate \(99,999^2\) directly (which would be messy), let's recognize this as a difference of squares pattern.

The difference of squares pattern says: when we have \(a^2 - b^2\), we can rewrite it as \((a + b)(a - b)\)

In our case:

• a = 99,999
• b = 1

So \(99,999^2 - 1^2 = (99,999 + 1)(99,999 - 1) = (100,000)(99,998)\)

This is much easier to work with than calculating \(99,999^2\) directly!

Process Skill: SIMPLIFY - Using algebraic patterns to avoid complex calculations

3. Express numbers using powers of 10

Now let's rewrite our numbers using powers of 10 to match the answer choice format:

\(100,000 = 10^5\) (that's 1 followed by 5 zeros)

\(99,998 = 100,000 - 2 = 10^5 - 2\)

So our expression becomes:

\(99,999^2 - 1^2 = (10^5)(10^5 - 2)\)

This matches the structure we see in the answer choices!

4. Apply the difference of squares formula

Let's put it all together:

Starting with: \(99,999^2 - 1^2\)
Using difference of squares: \((99,999 + 1)(99,999 - 1)\)
Simplifying: \((100,000)(99,998)\)
Expressing in powers of 10: \((10^5)(10^5 - 2)\)

This gives us: \(10^5(10^5 - 2)\)

4. Final Answer

Our result is \(10^5(10^5 - 2)\), which exactly matches answer choice E.

Let's verify this makes sense: we started with a number very close to 100,000 (which is \(10^5\)), squared it, and subtracted 1. Our factored form shows this relationship clearly.

The answer is E.

Common Faltering Points

Errors while devising the approach

1. Attempting direct calculation instead of recognizing algebraic patterns

Many students see \(99,999^2 - 1^2\) and immediately try to calculate \(99,999^2\) directly (which gives 9,999,800,001), then subtract 1. This leads to a messy numerical answer that doesn't match any of the given choices. Students fail to recognize that the answer choices contain expressions like \(10^5(10^5 - 2)\), which signals that an algebraic approach using difference of squares is needed.

2. Missing the connection between 99,999 and powers of 10

Students may not immediately see that \(99,999 = 100,000 - 1 = 10^5 - 1\). This connection is crucial for expressing the final answer in terms of powers of 10. Without this insight, students struggle to match their work to the answer choices, even if they apply the difference of squares correctly.

Errors while executing the approach

1. Arithmetic errors when applying difference of squares

When using \(a^2 - b^2 = (a + b)(a - b)\), students may incorrectly calculate:

• 99,999 + 1 as something other than 100,000
• 99,999 - 1 as something other than 99,998

These simple arithmetic mistakes derail the entire solution process.

2. Incorrect conversion to powers of 10

Even after getting \((100,000)(99,998)\), students may:

• Write 100,000 as \(10^4\) instead of \(10^5\) (counting zeros incorrectly)
• Express 99,998 as \(10^4 - 2\) instead of \(10^5 - 2\)
• Fail to recognize that \(99,998 = 100,000 - 2 = 10^5 - 2\)

Errors while selecting the answer

1. Confusing similar-looking answer choices

The answer choices contain very similar expressions:

• Choice C: \(10^4(10^5 - 2)\)
• Choice D: \(10^5(10^4 - 2)\)
• Choice E: \(10^5(10^5 - 2)\)

Students who correctly identify the need for \(10^5\) but make errors in the second factor might select choice C or D instead of the correct choice E. They may not carefully verify which specific combination of powers of 10 matches their algebraic work.