Which of the following is equivalent to 2^(p)*4^(p) + 1 ?

1. Translate the problem requirements: We need to find an equivalent expression for \(2^p \times 4^{p+1}\) by simplifying it into one of the given answer choices. This means combining the terms using exponent rules.
2. Convert to a common base: Since \(4 = 2^2\), we can rewrite everything in terms of base 2 to make the multiplication easier to handle.
3. Apply exponent multiplication rules: Use the rule that when multiplying terms with the same base, we add the exponents to combine into a single expression.
4. Match with answer choices: Compare our simplified result with the given options to identify the equivalent expression.

Execution of Strategic Approach

1. Translate the problem requirements

We have the expression \(2^p \times 4^{p+1}\) and we need to find which of the given answer choices is equivalent to it. Think of this like having two different types of building blocks that we want to combine into one single, simpler expression.

The key insight is that we're looking for an equivalent form - meaning when we plug in any value for p, both our original expression and the correct answer choice should give us the same numerical result.

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical goal

2. Convert to a common base

Right now we have \(2^p\) multiplied by \(4^{p+1}\). The challenge is that we have two different bases: 2 and 4. It's like trying to add apples and oranges - we need to express everything in the same "currency" first.

Let's think about what 4 really is: \(4 = 2 \times 2 = 2^2\)

So we can rewrite \(4^{p+1}\) as \((2^2)^{p+1}\).

Using the power rule for exponents, when we have \((a^m)^n\), this equals \(a^{m\times n}\). So:
\((2^2)^{p+1} = 2^{2\times(p+1)} = 2^{2p+2}\)

Now our original expression becomes: \(2^p \times 2^{2p+2}\)

Much better! Now we have the same base (2) for both terms.

3. Apply exponent multiplication rules

When we multiply terms that have the same base, we simply add the exponents together. Think of it this way: \(2^3 \times 2^4\) means \((2\times2\times2) \times (2\times2\times2\times2)\), which gives us \(2^7\).

So: \(2^p \times 2^{2p+2} = 2^{p + (2p+2)}\)

Let's combine the exponents step by step:
\(p + (2p + 2) = p + 2p + 2 = 3p + 2\)

Therefore: \(2^p \times 4^{p+1} = 2^{3p+2}\)

Process Skill: MANIPULATE - Systematically applying exponent rules to combine terms

4. Match with answer choices

Our simplified result is \(2^{3p+2}\).

Looking at the answer choices:

1. \(2^{3p+2}\) ✓ This matches exactly!
2. \(4^{2p}\)
3. \(8^{2p-2}\)
4. \(8^{2p+1}\)
5. \(8^{p(p+1)}\)

Let's verify this is correct by testing with a simple value. If \(p = 1\):

• Original: \(2^1 \times 4^{1+1} = 2 \times 4^2 = 2 \times 16 = 32\)
• Our answer: \(2^{3(1)+2} = 2^5 = 32\) ✓

Final Answer

The expression \(2^p \times 4^{p+1}\) is equivalent to \(2^{3p+2}\).

Answer: A

Common Faltering Points

Errors while devising the approach

1. Not recognizing the need for a common base: Students often try to work directly with different bases (2 and 4) without realizing they need to convert everything to the same base first. They might attempt to multiply \(2^p\) and \(4^{p+1}\) directly, leading to confusion about how to combine the terms.

2. Misunderstanding what "equivalent" means: Some students may think they need to solve for p or find a specific numerical value, rather than understanding that they need to algebraically manipulate the expression into a form that matches one of the answer choices.

Errors while executing the approach

1. Incorrect application of the power rule: When converting \(4^{p+1}\) to \((2^2)^{p+1}\), students frequently make the error of writing \(2^{2+p+1}\) instead of correctly applying the power rule to get \(2^{2(p+1)} = 2^{2p+2}\). This is a critical step that affects the entire solution.

2. Algebra mistakes when combining exponents: When adding the exponents \(p + (2p + 2)\), students may make simple arithmetic errors like forgetting to distribute properly or making sign errors, potentially getting results like \(2^{2p+2}\) or \(2^{3p+1}\) instead of the correct \(2^{3p+2}\).

3. Confusion with exponent multiplication vs. addition rules: Students sometimes confuse when to add exponents (same base multiplication) versus when to multiply exponents (power of a power), leading to incorrect manipulations throughout the solution process.

Errors while selecting the answer

1. Selecting answers with different bases without verification: Even after correctly deriving \(2^{3p+2}\), students might be tempted to select answer choices B, C, or D (which have bases 4 or 8) thinking they look more complex or "correct" without actually checking if they're equivalent to their derived answer.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for p

Let's use \(p = 1\) (simple value that makes calculations manageable)

Step 2: Calculate the original expression with p = 1

\(2^p \times 4^{p+1} = 2^1 \times 4^{1+1} = 2^1 \times 4^2 = 2 \times 16 = 32\)

Step 3: Test each answer choice with p = 1

1. \(2^{3p+2} = 2^{3(1)+2} = 2^5 = 32\) ✓
2. \(4^{2p} = 4^{2(1)} = 4^2 = 16\) ✗
3. \(8^{2p-2} = 8^{2(1)-2} = 8^0 = 1\) ✗
4. \(8^{2p+1} = 8^{2(1)+1} = 8^3 = 512\) ✗
5. \(8^{p(p+1)} = 8^{1(1+1)} = 8^2 = 64\) ✗

Step 4: Verify with another value

Let's check with \(p = 2\) to confirm:

Original: \(2^2 \times 4^{2+1} = 4 \times 4^3 = 4 \times 64 = 256\)

Choice A: \(2^{3(2)+2} = 2^8 = 256\) ✓

Conclusion: Only choice A gives the same result as the original expression for both test values, confirming it's the correct answer.