Problem 26 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \sqrt{\frac{1}{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Write the expression given

We have an expression given as: $$ \sqrt{\frac{1}{2}} $$ We will work on simplifying this expression.

Step 2: Rewrite the square root of a fraction as the product of the square roots of numerator and denominator

We can rewrite the square root of a fraction as the product of square roots of the numerator and the denominator. Therefore, we can rewrite the given expression as: $$ \sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} $$

Step 3: Simplify the square root of the numerator and denominator

Now, simplify the square root of the numerator (1) and the denominator (2): $$ \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} $$ Since the square root of 1 is 1, we don't need to simplify it further.

Step 4: Rationalize the denominator

To simplify the expression further, we'll rationalize the denominator (get rid of the square root in the denominator) by multiplying both numerator and denominator by \(\sqrt{2}\). So, we get: $$ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$ The denominator becomes 2 when we multiply \(\sqrt{2}\) by \(\sqrt{2}\). Now, the expression is simplified as: $$ \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2} $$

Key Concepts

Square Root

The square root symbol, \( \sqrt{} \), is used to find a number that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because \(2 \times 2 = 4\). The square root operation can also be applied to fractions.

When dealing with fractions under a square root, like \( \sqrt{\frac{1}{2}} \), it's often helpful to separate the numerator and the denominator. This means rewriting it as \( \sqrt{1} \) over \( \sqrt{2} \). This step simplifies the process and makes it easier to deal with each part individually.

Simplifying \( \sqrt{1} \) is straightforward since it equals 1. The challenge often comes from the denominator. The concept of square roots is essential in solving and understanding radical expressions.

Rationalizing the Denominator

Rationalizing the denominator involves removing any square root from the bottom of a fraction. This step is important because it allows for a more standard form and helps in comparing expressions.

To rationalize a denominator like \( \frac{1}{\sqrt{2}} \), you need to multiply both the numerator and the denominator by \( \sqrt{2} \). This process removes the square root in the denominator since \( \sqrt{2} \times \sqrt{2} = 2 \).

After performing this multiplication, the expression becomes \( \frac{\sqrt{2}}{2} \). This form is typically easier to work with and interpret. Rationalizing might seem a bit tedious, but it's a key skill in algebra that allows for expressions to be simplified in a standard way.

Radical Expressions

Radical expressions involve the use of roots, such as square roots, cubic roots, etc. They can sometimes seem daunting, but breaking them down into smaller parts makes them easier to manage.

With square roots, as seen in our example \( \sqrt{\frac{1}{2}} \), expressing the fraction in terms of its separate root parts \( \frac{\sqrt{1}}{\sqrt{2}} \) sets the stage for further simplification.

The goal with radical expressions is often to simplify them as much as possible for ease of use in further calculations. Simplifying, like turning \( \sqrt{\frac{1}{2}} \) into \( \frac{\sqrt{2}}{2} \), helps make problems involving radicals more straightforward and less error-prone. Mastering these techniques can greatly aid in more advanced mathematics.