- Translate the problem requirements: We're given \(\mathrm{x}\) equals the square root of a fraction \(\left(\frac{25}{16}\right)\), and we need to find the value of the square root of another fraction \(\left(\frac{5}{\mathrm{x}}\right)\). This requires finding \(\mathrm{x}\) first, then substituting it into the second expression.
- Simplify the given expression for x: Calculate \(\mathrm{x}\) by recognizing that \(\sqrt{\frac{25}{16}}\) can be simplified using the property that \(\sqrt{\frac{\mathrm{a}}{\mathrm{b}}} = \frac{\sqrt{\mathrm{a}}}{\sqrt{\mathrm{b}}}\), leading to a simple fraction.
- Substitute x into the target expression: Replace \(\mathrm{x}\) in \(\sqrt{\frac{5}{\mathrm{x}}}\) with the value found in the previous phase to create a new radical expression.
- Simplify the final radical expression: Use radical properties to simplify \(\sqrt{\frac{5}{\mathrm{x}}}\) into a clean numerical answer that matches one of the given choices.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we're asked to find. We have two parts to work with:
- First, we're given that \(\mathrm{x} = \sqrt{\frac{25}{16}}\)
- Second, we need to find the value of \(\sqrt{\frac{5}{\mathrm{x}}}\)
This means we need to find the value of \(\mathrm{x}\) first, then substitute that value into the second expression to get our final answer. Think of it like a two-step recipe - we can't make the final dish until we prepare the first ingredient.
Process Skill: TRANSLATE - Converting the problem statement into clear mathematical steps
2. Simplify the given expression for x
Now let's find the value of \(\mathrm{x} = \sqrt{\frac{25}{16}}\).
When we have the square root of a fraction, we can separate it into the square root of the top divided by the square root of the bottom. So \(\sqrt{\frac{25}{16}}\) becomes \(\sqrt{25} \div \sqrt{16}\).
Let's think about this step by step:
- \(\sqrt{25} = 5\) (because \(5 \times 5 = 25\))
- \(\sqrt{16} = 4\) (because \(4 \times 4 = 16\))
So \(\mathrm{x} = \sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}\)
Therefore: \(\mathrm{x} = \frac{5}{4}\)
3. Substitute x into the target expression
Now we need to find \(\sqrt{\frac{5}{\mathrm{x}}}\). We know that \(\mathrm{x} = \frac{5}{4}\), so let's substitute this value:
\(\sqrt{\frac{5}{\mathrm{x}}} = \sqrt{5 \div \frac{5}{4}}\)
When we divide by a fraction, we multiply by its reciprocal (flip it upside down). So:
\(5 \div \frac{5}{4} = 5 \times \frac{4}{5} = \frac{20}{5} = 4\)
So we need to find \(\sqrt{\frac{5}{\mathrm{x}}} = \sqrt{4}\)
4. Simplify the final radical expression
Now we just need to simplify \(\sqrt{4}\).
Since \(2 \times 2 = 4\), we know that \(\sqrt{4} = 2\).
Therefore: \(\sqrt{\frac{5}{\mathrm{x}}} = 2\)
Final Answer
The value of \(\sqrt{\frac{5}{\mathrm{x}}}\) is \(2\), which corresponds to answer choice C.
Let's verify: We found \(\mathrm{x} = \frac{5}{4}\), then \(\sqrt{\frac{5}{\mathrm{x}}} = \sqrt{5 \div \frac{5}{4}} = \sqrt{5 \times \frac{4}{5}} = \sqrt{4} = 2\) ✓
Common Faltering Points
Errors while devising the approach
- Misunderstanding the nested structure: Students may try to solve for \(\sqrt{\frac{5}{\mathrm{x}}}\) directly without first finding the value of \(\mathrm{x}\). They might attempt to substitute the entire expression \(\sqrt{\frac{25}{16}}\) into the second expression, leading to a complex nested radical that's much harder to simplify.
- Confusion about order of operations: Students might think they need to work from the inside out of \(\sqrt{\frac{5}{\mathrm{x}}}\) first, trying to simplify \(\frac{5}{\mathrm{x}}\) before knowing what \(\mathrm{x}\) equals, rather than recognizing they must find \(\mathrm{x}\) first as a separate step.
Errors while executing the approach
- Fraction division errors: When calculating \(5 \div \left(\frac{5}{4}\right)\), students commonly forget to multiply by the reciprocal. They might incorrectly compute this as \(\frac{5 \times 5}{4} = \frac{25}{4}\) instead of the correct \(5 \times \left(\frac{4}{5}\right) = 4\).
- Radical simplification mistakes: Students may incorrectly separate \(\sqrt{\frac{25}{16}}\) as \(\frac{\sqrt{25}}{\sqrt{16}}\) but then make arithmetic errors, getting \(\sqrt{25} = 25\) or \(\sqrt{16} = 16\) instead of the correct values 5 and 4 respectively.
- Sign confusion with square roots: Some students might consider both positive and negative values when taking square roots (like \(\pm 5\) for \(\sqrt{25}\)), not recognizing that in this context we use the principal (positive) square root.
Errors while selecting the answer
- Selecting an intermediate result: Students might correctly find that \(\mathrm{x} = \frac{5}{4}\) but then mistakenly select answer choice B \(\left(\frac{1}{4}\right)\) or confuse this with the reciprocal \(\frac{4}{5}\), rather than continuing to find the final answer \(\sqrt{\frac{5}{\mathrm{x}}} = 2\).
