Which of the following is equal to (12 + sqrt(28))/(sqrt(9 + 7))?

1. Translate the problem requirements: We need to simplify the expression \(\frac{12 + \sqrt{28}}{\sqrt{9 + 7}}\) and match it to one of the given answer choices.
2. Simplify the radical expressions: Break down \(\sqrt{28}\) and \(\sqrt{9 + 7}\) into their simplest forms by identifying perfect square factors.
3. Rationalize the denominator: Since we have a square root in the denominator, multiply both numerator and denominator by the conjugate to eliminate the radical from the denominator.
4. Simplify the resulting expression: Perform the multiplication and combine like terms to arrive at the final simplified form that matches one of the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we need to do. We have the expression \(\frac{12 + \sqrt{28}}{\sqrt{9 + 7}}\) and we need to simplify it to match one of the answer choices.

First, let's look at what we have in plain terms: we have a fraction where the top part (numerator) contains the number 12 plus the square root of 28, and the bottom part (denominator) is the square root of the sum 9 plus 7.

Process Skill: TRANSLATE - Converting the mathematical expression into clear understanding of what needs to be simplified

2. Simplify the radical expressions

Now let's work on making those square roots simpler.

For the denominator: \(\sqrt{9 + 7} = \sqrt{16} = 4\)
This is straightforward - 9 plus 7 equals 16, and the square root of 16 is 4.

For the numerator, we need to simplify \(\sqrt{28}\). Let's think about what factors 28 has:
28 = 4 × 7 = 2² × 7

Since 4 is a perfect square (2²), we can take it out of the square root:
\(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\)

So our expression becomes:
\(\frac{12 + 2\sqrt{7}}{4}\)

Process Skill: SIMPLIFY - Breaking down complex radicals into their simplest forms

3. Rationalize the denominator

Actually, looking at our current expression \(\frac{12 + 2\sqrt{7}}{4}\), we notice that the denominator is already a simple number (4), not a radical. So we don't need to rationalize in the traditional sense - we just need to simplify this fraction.

4. Simplify the resulting expression

Now we can divide both terms in the numerator by 4:

\(\frac{12 + 2\sqrt{7}}{4} = \frac{12}{4} + \frac{2\sqrt{7}}{4}\)

Simplifying each part:

• \(\frac{12}{4} = 3\)
• \(\frac{2\sqrt{7}}{4} = \frac{\sqrt{7}}{2}\)

Therefore: \(\frac{12 + 2\sqrt{7}}{4} = 3 + \frac{\sqrt{7}}{2}\)

Final Answer

Our simplified expression is \(3 + \frac{\sqrt{7}}{2}\), which matches answer choice (D): \(3 + \frac{\sqrt{7}}{2}\).

Let's verify: Starting with \(\frac{12 + \sqrt{28}}{\sqrt{9 + 7}}\), we simplified to \(\frac{12 + 2\sqrt{7}}{4}\), which equals \(3 + \frac{\sqrt{7}}{2}\).

The answer is D.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the order of operations in the denominator: Students may try to simplify \(\sqrt{9} + \sqrt{7}\) instead of \(\sqrt{9 + 7}\). This is a critical error because \(\sqrt{a + b} ≠ \sqrt{a} + \sqrt{b}\). The expression clearly shows 9 + 7 inside the square root, which must be calculated first before taking the square root.

2. Overlooking the need to simplify \(\sqrt{28}\): Students might leave \(\sqrt{28}\) as is without recognizing that it can be simplified. This prevents them from seeing that 28 = 4 × 7, where 4 is a perfect square that can be extracted, leading to \(2\sqrt{7}\).

Errors while executing the approach

1. Incorrect factorization of 28: When simplifying \(\sqrt{28}\), students may factor incorrectly (such as 28 = 14 × 2 instead of 4 × 7) or fail to identify that 4 is the perfect square factor. This leads to an incorrect simplification like \(\sqrt{14} \cdot \sqrt{2}\) instead of \(2\sqrt{7}\).

2. Arithmetic errors when dividing by 4: In the final step \(\frac{12 + 2\sqrt{7}}{4}\), students may incorrectly calculate \(\frac{12}{4}\) or \(\frac{2\sqrt{7}}{4}\). Common mistakes include getting 4 instead of 3, or \(\sqrt{7}\) instead of \(\frac{\sqrt{7}}{2}\).

3. Attempting unnecessary rationalization: Students may try to rationalize the denominator even when it's already a rational number (4), leading them down an incorrect path and potentially multiplying by conjugates unnecessarily.

Errors while selecting the answer

1. Selecting the intermediate step instead of the final answer: Students may arrive at the correct intermediate expression \(\frac{12 + 2\sqrt{7}}{4}\) but then select answer choice A, which matches this form, instead of continuing to simplify to get the final answer \(3 + \frac{\sqrt{7}}{2}\).