Problem 49 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{\frac{1}{2}}\) is \(\frac{\sqrt{2}}{2}\).

Step 1: Recognize the Form

Recognize that \(\sqrt{\frac{1}{2}}\) means square root of \(1/2\). This can be simplified into the form \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), where \(a = 1\) and \(b = 2\).

Step 2: Simplify the Roots

We can simplify the roots separately. The square root of \(1\) is simply \(1\), and the square root of \(2\) is \(\sqrt{2}\). In a simplified form, this yields the expression \(\frac{1}{\sqrt{2}}\).

Step 3: Rationalize the Denominator

Rationalize the denominator by multiplying the numerator and the denominator by \(\sqrt{2}\). Therefore, our final simplified expression is \(\frac{\sqrt{2}}{2}\).

Key Concepts

Rationalizing the Denominator

When dealing with radicals, especially fractions that involve square roots in the denominator, it's often necessary to "rationalize the denominator". This process eliminates the square root from the bottom part of the fraction. The main goal is to convert the denominator into a rational number, which is a number without a square root.

To rationalize the denominator, multiply both the numerator and the denominator by the same square root present in the denominator. For instance, if we have \(\frac{1}{\sqrt{2}}\), multiplying the numerator and denominator by \(\sqrt{2}\) gives us \(\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}\), transforming it into \(\frac{\sqrt{2}}{2}\).

Rationalizing is a useful technique because it simplifies expressions, particularly making them easier to handle in more complex algebraic operations or when you aim for neat numeric expressions.

Square Roots

The square root operation is fundamental when simplifying expressions that contain radicals. A square root is essentially a number which produces a specific quantity when multiplied by itself. For example, the square root of 4 is 2, since \(2 \times 2 = 4\). Similarly, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

When you see \(\sqrt{\frac{1}{2}}\), it signifies the square root of the fraction \(\frac{1}{2}\). To approach this, split the radicals: apply the square root to the numerator and the square root to the denominator individually. Thus, \(\sqrt{\frac{1}{2}}\) becomes \(\frac{\sqrt{1}}{\sqrt{2}}\). Since \(\sqrt{1} = 1\), this simplifies the whole expression to \(\frac{1}{\sqrt{2}}\).

Understanding how to separate the square root over a fraction is key to simplifying and solving radical expressions efficiently.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. It makes fractions easier to read, understand, and work with mathematically. When simplifying fractions that include radicals, rationalizing the denominator often forms part of this process, as we've seen.

Take a fraction like \(\frac{2}{4}\); you can simplify it by dividing both the numerator and the denominator by the greatest common divisor, which in this case is 2, giving \(\frac{1}{2}\).

However, when radicals are involved, simplification includes other steps, such as ensuring there are no square roots in the denominator, as in the example of \(\frac{1}{\sqrt{2}}\). Once you rationalize and simplify, you achieve a reduced fraction form, like \(\frac{\sqrt{2}}{2}\). This process not only simplifies the expression but also aids in clearer mathematical communication.