Which of the following is equal to \(2^k \times 5^{(k-1)}\)?

1. Translate the problem requirements: We need to rewrite the expression \(\mathrm{2^k \times 5^{(k-1)}}\) in a form that matches one of the given answer choices, all of which involve powers of 10.
2. Recognize the connection to powers of 10: Since \(\mathrm{10 = 2 \times 5}\), we should look for ways to create factors of 10 from the given expression.
3. Factor strategically to create powers of 10: Separate out one factor of 2 from \(\mathrm{2^{(k-1)}}\) and combine the remaining \(\mathrm{2^{(k-1)}}\) with \(\mathrm{5^{(k-1)}}\) to form \(\mathrm{10^{(k-1)}}\).
4. Match with answer choices: Compare the resulting expression with the given options to identify the correct answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we need to do. We have the expression \(\mathrm{2^k \times 5^{(k-1)}}\) and we need to rewrite it in a different form. Looking at the answer choices, they all involve powers of 10 - like \(\mathrm{10^k}\) or \(\mathrm{10^{(k-1)}}\). This tells us we need to find a way to express our original expression using powers of 10 instead of separate powers of 2 and 5.

The key insight is recognizing that \(\mathrm{10 = 2 \times 5}\), so powers of 10 can be broken down into powers of 2 and 5, and vice versa.

Process Skill: TRANSLATE - Converting the algebraic expression into a form that matches the answer format

2. Recognize the connection to powers of 10

Since we know that \(\mathrm{10 = 2 \times 5}\), we can work backwards. If we see \(\mathrm{10^n}\), that's the same as \(\mathrm{(2 \times 5)^n = 2^n \times 5^n}\).

In our expression \(\mathrm{2^k \times 5^{(k-1)}}\), we have different exponents on the 2 and the 5. The power of 2 is k, while the power of 5 is (k-1). To create a power of 10, we need the same exponent on both 2 and 5.

Since 5 has the smaller exponent (k-1), let's see if we can create \(\mathrm{10^{(k-1)}}\) by using \(\mathrm{2^{(k-1)} \times 5^{(k-1)}}\).

3. Factor strategically to create powers of 10

Let's manipulate our expression step by step. We start with:
\(\mathrm{2^k \times 5^{(k-1)}}\)

We can rewrite \(\mathrm{2^k}\) as \(\mathrm{2^1 \times 2^{(k-1)}}\), which gives us:
\(\mathrm{2^1 \times 2^{(k-1)} \times 5^{(k-1)}}\)

Now we can group the terms with the same exponent:
\(\mathrm{2^1 \times [2^{(k-1)} \times 5^{(k-1)}]}\)

The expression in brackets can be rewritten using the rule that \(\mathrm{a^n \times b^n = (a \times b)^n}\):
\(\mathrm{2^1 \times [2 \times 5]^{(k-1)}}\)

Since \(\mathrm{2 \times 5 = 10}\):
\(\mathrm{2^1 \times 10^{(k-1)}}\)

Which simplifies to:
\(\mathrm{2 \times 10^{(k-1)}}\)

Process Skill: MANIPULATE - Using exponent rules to strategically regroup terms

4. Match with answer choices

Our final expression is \(\mathrm{2 \times 10^{(k-1)}}\).

Looking at the answer choices:

1. \(\mathrm{2 \times 10^{(k-1)}}\) ✓ This matches exactly!
2. \(\mathrm{5 \times 10^{(k-1)}}\) ✗
3. \(\mathrm{10^k}\) ✗
4. \(\mathrm{2 \times 10^k}\) ✗
5. \(\mathrm{10^{(2k-1)}}\) ✗

The answer is A.

We can verify this works by expanding our answer back: \(\mathrm{2 \times 10^{(k-1)} = 2 \times [2 \times 5]^{(k-1)} = 2 \times 2^{(k-1)} \times 5^{(k-1)} = 2^k \times 5^{(k-1)}}\) ✓

Common Faltering Points

Errors while devising the approach

1. Not recognizing the need to use powers of 10
Students may try to simplify \(\mathrm{2^k \times 5^{(k-1)}}\) using other algebraic manipulations without realizing that all answer choices involve powers of 10. This means they need to express their answer using \(\mathrm{10^n}\) format, which requires understanding that \(\mathrm{10 = 2 \times 5}\).

2. Attempting to directly combine unlike exponents
Students might try to incorrectly combine \(\mathrm{2^k \times 5^{(k-1)}}\) as \(\mathrm{(2\times5)^k}\) or \(\mathrm{(2\times5)^{(k-1)}}\) without recognizing that the exponents are different (k vs k-1), so they cannot directly use the rule \(\mathrm{a^n \times b^n = (ab)^n}\).

Errors while executing the approach

1. Incorrect application of exponent rules when factoring
When rewriting \(\mathrm{2^k}\) as \(\mathrm{2^1 \times 2^{(k-1)}}\), students might make errors such as writing \(\mathrm{2^k = 2^k \times 2^{(k-1)}}\) (adding exponents incorrectly) instead of correctly using the rule that \(\mathrm{2^k = 2^1 \times 2^{(k-1)}}\) since \(\mathrm{1 + (k-1) = k}\).

2. Mixing up which terms to group together
Students may incorrectly group terms when trying to create powers of 10. For example, they might try to group \(\mathrm{2^k}\) with \(\mathrm{5^{(k-1)}}\) directly, or factor out the wrong power, leading to expressions like \(\mathrm{5 \times 10^{(k-1)}}\) instead of the correct \(\mathrm{2 \times 10^{(k-1)}}\).

Errors while selecting the answer

1. Confusing similar-looking answer choices
Students may correctly arrive at an expression involving 2 and \(\mathrm{10^{(k-1)}}\) but then select answer choice D (\(\mathrm{2 \times 10^k}\)) instead of the correct answer A (\(\mathrm{2 \times 10^{(k-1)}}\)), mixing up the exponents k and (k-1) in the final selection step.