If x > 0, which of the following is equal to sqrt(32x^5)?

1. Translate the problem requirements: We need to simplify \(\sqrt{32x^5}\) where \(x > 0\), which means finding an equivalent expression by extracting perfect squares from under the radical sign.
2. Factor out perfect squares from the coefficient: Break down 32 into its prime factors to identify which parts can be taken out of the square root as whole numbers.
3. Handle the variable portion systematically: Deal with \(x^5\) by separating it into parts that form perfect squares (even powers) and parts that must remain under the radical.
4. Combine and verify against answer choices: Put together the simplified components and match with the given options to confirm our result.

Execution of Strategic Approach

1. Translate the problem requirements

We need to simplify \(\sqrt{32x^5}\) where \(x > 0\). Think of this like organizing a messy closet - we want to take out anything that comes in complete pairs and leave behind only the single items that don't have matches.

In square root terms, this means we're looking for pieces under the radical sign that are "perfect squares" - numbers or variables that can be written as something multiplied by itself. When we find these perfect squares, we can pull them out from under the square root sign as their simpler form.

Since \(x > 0\), we don't need to worry about negative values, which makes our work cleaner.

Process Skill: TRANSLATE - Converting the square root simplification requirement into a systematic search for perfect square factors

2. Factor out perfect squares from the coefficient

Let's break down the number 32 into its basic building blocks (prime factors) to see what we can extract.

\(32 = 2 \times 16 = 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\)

Now, since we're dealing with a square root, we need pairs of identical factors. From \(2^5\), we can make two complete pairs of 2's, with one 2 left over:
\(2^5 = (2^2) \times (2^2) \times 2 = 4 \times 4 \times 2\)

The two 4's represent perfect squares that can come out of the radical. When we take \(\sqrt{4 \times 4}\), we get 4. The remaining 2 stays under the radical.

So from the coefficient: \(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\)

3. Handle the variable portion systematically

Now let's deal with \(x^5\) in the same organized way. We need to separate the variable part into pieces that form perfect squares (even powers) and pieces that must stay under the radical.

\(x^5 = x^4 \times x^1\)

Here's the key insight: \(x^4\) is a perfect square because \(x^4 = (x^2)^2\). When we take the square root of a perfect square, we get the base: \(\sqrt{x^4} = \sqrt{(x^2)^2} = x^2\)

The remaining \(x^1\) (just x) doesn't have a partner, so it stays under the radical.

So from the variable part: \(\sqrt{x^5} = \sqrt{x^4 \times x} = \sqrt{x^4} \times \sqrt{x} = x^2 \times \sqrt{x} = x^2\sqrt{x}\)

4. Combine and verify against answer choices

Now we put our simplified pieces together:

\(\sqrt{32x^5} = \sqrt{32} \times \sqrt{x^5} = 4\sqrt{2} \times x^2\sqrt{x} = 4x^2\sqrt{2x}\)

Let's verify this matches our answer choices:

1. \(2x\sqrt{8x^4}\) - This doesn't match our form
2. \(16x\sqrt{2x}\) - The coefficient and power are wrong
3. \(4x^2\sqrt{2x}\) - This exactly matches our result!
4. \(8x^2\sqrt{2x^2}\) - Wrong coefficient and what's under the radical
5. \(8x^4\sqrt{4x}\) - Wrong coefficient and powers

Process Skill: SIMPLIFY - Systematically extracting perfect square factors to reduce the complexity of the radical expression

Final Answer

The answer is C: \(4x^2\sqrt{2x}\)

Our systematic approach of separating perfect squares from non-perfect squares gave us exactly what we needed. We extracted 4 from the coefficient 32, pulled out \(x^2\) from \(x^5\), and left 2x under the radical sign.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "simplifying a square root" means: Students may think they need to calculate the decimal value of \(\sqrt{32x^5}\) rather than expressing it in simplified radical form. This leads them to attempt decimal approximations instead of factoring out perfect squares.

2. Overlooking the constraint \(x > 0\): Students might worry unnecessarily about absolute value signs or negative possibilities when dealing with even powers like \(x^4\), not realizing that the given constraint \(x > 0\) eliminates these concerns and allows for cleaner simplification.

3. Not recognizing the perfect square factorization strategy: Students may attempt to work with the expression \(\sqrt{32x^5}\) as a whole unit rather than understanding they need to systematically separate it into \(\sqrt{32} \times \sqrt{x^5}\) and then factor each part to extract perfect squares.

Errors while executing the approach

1. Incorrect prime factorization of 32: Students commonly make arithmetic errors when breaking down 32, such as writing \(32 = 2^4\) instead of \(32 = 2^5\), which leads to extracting the wrong coefficient (\(2^2 = 4\) instead of the correct 4 from \(\sqrt{2^4}\)).

2. Mishandling variable exponents in radicals: Students frequently confuse how to split \(x^5\) under a square root. Common errors include writing \(\sqrt{x^5} = x^{5/2}\) (correct but not in the required form) or incorrectly splitting it as \(x^3\sqrt{x^2}\) instead of the correct \(x^2\sqrt{x}\).

3. Calculation errors when combining terms: When combining \(4\sqrt{2}\) and \(x^2\sqrt{x}\), students may incorrectly write the final answer as \(4x^2\sqrt{2} \times \sqrt{x}\) instead of properly combining the radicals to get \(4x^2\sqrt{2x}\).

Errors while selecting the answer

1. Failing to verify the final form against answer choices: Students may arrive at equivalent expressions like \(4x^2\sqrt{2} \times \sqrt{x}\) but fail to combine the radicals properly for comparison with the given options, missing that this equals \(4x^2\sqrt{2x}\).

2. Confusing similar-looking answer choices: Students might select option D (\(8x^2\sqrt{2x^2}\)) thinking it's equivalent to their answer \(4x^2\sqrt{2x}\), not carefully checking that both the coefficient (8 vs 4) and the expression under the radical (\(2x^2\) vs \(2x\)) are different.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a strategic value for x

Since we need to simplify \(\sqrt{32x^5}\), let's choose \(x = 2\). This gives us a concrete expression to work with: \(\sqrt{32 \times 2^5} = \sqrt{32 \times 32} = \sqrt{1024}\).

Step 2: Calculate the simplified form

\(\sqrt{1024} = 32\) (since \(32^2 = 1024\))

Step 3: Test each answer choice with \(x = 2\)

A. \(2x\sqrt{8x^4} = 2(2)\sqrt{8 \times 16} = 4\sqrt{128} = 4 \times 8\sqrt{2} = 32\sqrt{2} \neq 32\)

B. \(16x\sqrt{2x} = 16(2)\sqrt{2 \times 2} = 32\sqrt{4} = 32 \times 2 = 64 \neq 32\)

C. \(4x^2\sqrt{2x} = 4(4)\sqrt{2 \times 2} = 16\sqrt{4} = 16 \times 2 = 32 ✓\)

D. \(8x^2\sqrt{2x^2} = 8(4)\sqrt{2 \times 4} = 32\sqrt{8} = 32 \times 2\sqrt{2} = 64\sqrt{2} \neq 32\)

E. \(8x^4\sqrt{4x} = 8(16)\sqrt{4 \times 2} = 128\sqrt{8} = 128 \times 2\sqrt{2} = 256\sqrt{2} \neq 32\)

Step 4: Verify with another value

Let's also test \(x = 1\) to confirm: \(\sqrt{32 \times 1^5} = \sqrt{32} = 4\sqrt{2}\)

Choice C: \(4(1)^2\sqrt{2 \times 1} = 4\sqrt{2} ✓\)

Conclusion: Only choice C consistently matches our calculated values, confirming it as the correct answer.