(1/(sqrt(2)))/(sqrt(3/2))

1. Translate the problem requirements: We need to simplify a complex fraction where the numerator is \(\frac{1}{\sqrt{2}}\) and the denominator is \(\sqrt{\frac{3}{2}}\). The goal is to express this as a single simplified radical expression.
2. Convert division to multiplication: Transform the complex fraction into a multiplication problem by taking the reciprocal of the denominator.
3. Simplify the square root in the denominator: Break down \(\sqrt{\frac{3}{2}}\) into a more manageable form using the property that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).
4. Perform the multiplication and rationalize: Multiply the fractions and eliminate any square roots from the denominator by rationalizing.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking at. We have a fraction where the top part (numerator) is itself a fraction: \(\frac{1}{\sqrt{2}}\). The bottom part (denominator) is a square root of a fraction: \(\sqrt{\frac{3}{2}}\).

Think of this like having a sandwich where the filling is also made of smaller sandwiches - we need to unwrap each layer carefully. Our goal is to simplify this complex expression into one clean, simple form that matches one of our answer choices.

Mathematically, we're working with: \(\left(\frac{1}{\sqrt{2}}\right) \div \sqrt{\frac{3}{2}}\)

Process Skill: TRANSLATE - Converting the complex fraction notation into a division problem we can work with

2. Convert division to multiplication

When we divide by something, it's the same as multiplying by its reciprocal (flipping it upside down). This is like saying "dividing by 1/2 is the same as multiplying by 2/1".

So our expression \(\left(\frac{1}{\sqrt{2}}\right) \div \sqrt{\frac{3}{2}}\) becomes: \(\left(\frac{1}{\sqrt{2}}\right) \times \frac{1}{\sqrt{\frac{3}{2}}}\)

This gives us: \(\frac{1}{\sqrt{2} \times \sqrt{\frac{3}{2}}}\)

3. Simplify the square root in the denominator

Now we need to work with \(\sqrt{\frac{3}{2}}\). Remember that the square root of a fraction equals the square root of the top divided by the square root of the bottom.

So \(\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}\)

Our expression becomes: \(\frac{1}{\sqrt{2} \times \frac{\sqrt{3}}{\sqrt{2}}}\)

In the denominator, we have \(\sqrt{2} \times \frac{\sqrt{3}}{\sqrt{2}}\). When we multiply \(\sqrt{2}\) by \(\frac{\sqrt{3}}{\sqrt{2}}\), the \(\sqrt{2}\) terms cancel out, leaving us with just \(\sqrt{3}\).

So our expression simplifies to: \(\frac{1}{\sqrt{3}}\)

4. Perform the multiplication and rationalize

We now have \(\frac{1}{\sqrt{3}}\), but we need to get rid of the square root in the denominator. This process is called rationalizing.

To rationalize \(\frac{1}{\sqrt{3}}\), we multiply both the top and bottom by \(\sqrt{3}\):

\(\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{3}}{3}\)

This works because \(\sqrt{3} \times \sqrt{3} = 3\), and we're essentially multiplying by 1 (since \(\frac{\sqrt{3}}{\sqrt{3}} = 1\)), so we're not changing the value.

Process Skill: MANIPULATE - Using rationalization to eliminate square roots from denominators

Final Answer

Our simplified expression is \(\frac{\sqrt{3}}{3}\), which matches answer choice B.

To verify: We started with \(\frac{\frac{1}{\sqrt{2}}}{\sqrt{\frac{3}{2}}}\) and through systematic simplification arrived at \(\frac{\sqrt{3}}{3}\). This represents the same value in its simplest rationalized form.

Common Faltering Points

Errors while devising the approach

Students often get confused by the nested fraction notation and may try to multiply all terms together instead of recognizing this as a division problem. They might see \(\frac{\frac{1}{\sqrt{2}}}{\sqrt{\frac{3}{2}}}\) and incorrectly think it means \((1)(\sqrt{2})(\sqrt{\frac{3}{2}})\) rather than understanding it's \(\left(\frac{1}{\sqrt{2}}\right) \div \sqrt{\frac{3}{2}}\).

Some students may attempt to rationalize denominators at each step individually rather than first converting the division to multiplication and simplifying. This leads to more complex intermediate steps and increases chances of arithmetic errors.

Errors while executing the approach

When simplifying \(\sqrt{\frac{3}{2}}\), students frequently make the error \(\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{2}\) instead of the correct \(\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}\). This fundamental mistake with square root of fractions leads to completely wrong final answers.

In the step where \(\sqrt{2} \times \frac{\sqrt{3}}{\sqrt{2}}\) should simplify to \(\sqrt{3}\), students may incorrectly cancel terms or make multiplication errors, such as getting \(\sqrt{6}\) instead of \(\sqrt{3}\), or failing to recognize that \(\frac{\sqrt{2}}{\sqrt{2}} = 1\).

After correctly arriving at \(\frac{1}{\sqrt{3}}\), many students forget that GMAT answers require rationalized denominators and select an answer choice that would correspond to \(\frac{1}{\sqrt{3}}\) if it were available, rather than completing the rationalization to get \(\frac{\sqrt{3}}{3}\).

Errors while selecting the answer

Students who arrive at \(\frac{1}{\sqrt{3}}\) may look at the answer choices and incorrectly match it with choice E (\(\sqrt{3}\)), thinking they're equivalent, rather than recognizing that their answer needs to be rationalized to \(\frac{\sqrt{3}}{3}\) which corresponds to choice B.