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Which of the following is equal to \(2^k \times 5^{(k-1)}\)?
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
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 \(\mathrm{2^{(k-1)} \times 5^{(k-1)}}\).
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
Our final expression is \(\mathrm{2 \times 10^{(k-1)}}\).
Looking at the answer choices:
The answer is A.
We can verify this works by expanding our answer back: \(\mathrm{2 \times 10^{(k-1)} = 2 \times [2 \times 5]^{(k-1)} = 2 \times 2^{(k-1)} \times 5^{(k-1)} = 2^k \times 5^{(k-1)}}\) ✓
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 \(\mathrm{a^n \times b^n = (ab)^n}\).
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 \(\mathrm{2 \times 10^{(k-1)}}\).
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.