Problem 33 - Technical Mathematics with Calculus

Question

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{\frac{1}{2 x}}$$

Step-by-Step Solution

Answer

\( \frac{1}{\sqrt{2x}} \)

Step 1: Understand the expression

The given expression is a square root of a fraction. In this expression, we have the square root of the reciprocal of a variable multiplied by 2. The expression is \(\sqrt{\frac{1}{2x}}\).

Step 2: Apply the property of radicals over fraction

The square root of a fraction can be split into the square root of the numerator and the square root of the denominator separately. This means \(\sqrt{\frac{1}{2x}} = \frac{\sqrt{1}}{\sqrt{2x}}\).

Step 3: Simplify the square root of the numerator

Since the square root of 1 is 1, the expression simplifies to \( \frac{1}{\sqrt{2x}} \).

Step 4: Rationalize the denominator

To remove the radical from the denominator, we multiply both the numerator and the denominator by the square root of the denominator. So we get \( \frac{1}{\sqrt{2x}}\times\frac{\sqrt{2x}}{\sqrt{2x}} = \frac{\sqrt{2x}}{2x} \).

Step 5: Simplify the result

After rationalization, we can see that the radical in the numerator and the variable in the denominator are the same term squared. Therefore, the expression simplifies to \( \frac{1}{\sqrt{2x}} \). This is the simplest form, and we do not simplify any further because the expression inside the radical is already in its simplest form, not having any perfect squares.

Key Concepts

Square Root Simplification

Simplifying square roots involves finding an equivalent expression where the radical is as simple as possible. To grasp this concept, imagine peeling layers off an onion to get to the core—the simplest form of the number or expression under the radical sign.

Starting with, a number like \( \sqrt{16} \) can be simplified to \( 4 \) because 4 is the number that, when squared, gives 16. Simplification becomes a bit trickier with variables and fractions. For instance, the square root of a fraction like \( \frac{1}{2x} \) requires a methodical approach:

• Separate the radical over the numerator and the denominator as \( \frac{\sqrt{1}}{\sqrt{2x}} \).


• Determine the square root of the numerator. Since \( \sqrt{1} \) is 1, this simplifies to \( \frac{1}{\sqrt{2x}} \).

As easy as it might seem, the process gets complex with non-perfect squares and variable expressions. For these, keeping the expression under the radical as simplified as possible, without any perfect square factors, is the key objective.

Rationalizing the Denominator

Rationalizing the denominator revolves around the principle of not leaving a radical in the denominator of a fraction. To understand why this matters, consider the convenience it provides when adding, subtracting, or comparing fractions with radicals.

To rationalize a denominator containing a radical, you multiply the fraction by a form of 1 that will eliminate the radical in the denominator, typically using the conjugate or the radical itself. For example:

• To rationalize \( \frac{1}{\sqrt{2x}} \) we multiply by \( \frac{\sqrt{2x}}{\sqrt{2x}} \) - essentially multiplying by 1 to maintain equality.


• This results in \( \frac{\sqrt{2x}}{2x} \), as seen in our exercise.

The choice of \( \sqrt{2x} \) as the multiplier is purposeful: it’s the simplest form that, when multiplied by itself (\( \sqrt{2x}\times\sqrt{2x} \) ), gives a rational number ( \( 2x \) ). By doing so, you're ensuring that the denominator can be simplified to a non-radical expression, thereby 'rationalizing' it.

Radicals Over Fractions

Handling radicals over fractions––or fractional radicands––involves separate consideration of the numerator and the denominator under the radical sign. The process allows for a clearer path to simplification and potential rationalization later on.

The steps to simplify such expressions are:

• Split the radical between the numerator and the denominator as \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).


• Simplify the resulting radicals, if possible.


• Finally, if you’re left with a radical in the denominator, you’d proceed to rationalize it as discussed above.

This process not only simplifies calculation but also paves the way for a standardized form, making it easier to perform operations with other radicals or fractions. It's essential to handle each part—the numerator and the denominator—carefully to ensure a fully simplified and rationalized result.