Problem 31 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{0.5} $$

Step-by-Step Solution

Answer

The simplest radical form of \( \sqrt{0.5} \) is \( \frac{\sqrt{2}}{2} \).

Step 1: Express the Radical as a Fraction

To simplify \( \sqrt{0.5} \), first rewrite the decimal as a fraction: \( 0.5 = \frac{1}{2} \).

Step 2: Rewrite the Expression as a Radical Fraction

Now express the square root of the fraction: \( \sqrt{\frac{1}{2}} \). This expression can be split into the square roots of the numerator and the denominator: \( \frac{\sqrt{1}}{\sqrt{2}} \).

Step 3: Simplify the Numerator

The square root of the numerator, \( \sqrt{1} \), simplifies directly to 1, since \( 1^2 = 1 \).

Step 4: Rationalize the Denominator

Since the denominator contains a square root, we rationalize it by multiplying the numerator and the denominator by \( \sqrt{2} \). This gives: \( \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2} \).

Step 5: Final Simplified Form

The simplest form of \( \sqrt{0.5} \) is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate square roots or other radicals from the bottom (denominator) of a fraction. The idea is to make the denominator a rational number, meaning without any radical. This makes the expression cleaner and easier to work with. To do this, you multiply both the numerator and the denominator by the same radical that will cancel out the one in the denominator. For example, when you have an expression like \( \frac{1}{\sqrt{2}} \), you multiply both top and bottom by \( \sqrt{2} \) because \( \sqrt{2} \cdot \sqrt{2} = 2 \), thus getting:

• Multiplication: \( \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2} \)

The radical cancels out, leaving a rational number in the denominator and simplifying the overall expression. This step helps in expressing radicals in their simplest form, as shown in the solution where \( \sqrt{0.5} \) is rationalized to \( \frac{\sqrt{2}}{2} \).

Square Root

The square root is a special number that, when multiplied by itself, gives the original number. It is represented using the radical symbol \( \sqrt{} \). Understanding square roots involves identifying perfect squares and simplifying roots of non-perfect squares. For instance:

• The square root of 1 is 1 because \( 1 \times 1 = 1 \).
• \( \sqrt{2} \) remains as it is, since 2 is not a perfect square.

When you encounter a square root of a fraction, like \( \sqrt{\frac{1}{2}} \), you can separate it into the square root of the numerator and the square root of the denominator. This is

• Step-by-Step: \( \frac{\sqrt{1}}{\sqrt{2}} \) simplifies to \( \frac{1}{\sqrt{2}} \), demonstrating how to work with radical fractions efficiently.

Simplifying roots and understanding how they function in different scenarios leads to a deeper grasp of mathematical concepts and helps in tackling various problems effectively.

Fraction to Decimal Conversion

Fraction to decimal conversion is an important concept when simplifying expressions. Since some students may struggle with moving between the two, understanding how to convert effectively is key. This process involves dividing the numerator by the denominator in a fraction to get a decimal number. For instance:

• \( \frac{1}{2} \) can be divided—1 divided by 2 equals 0.5.

This skill is essential when translating numbers between formats in different problems. For example, the initial step in simplifying \( \sqrt{0.5} \) was recognizing 0.5 as a fraction, \( \frac{1}{2} \). Moving back and forth between these forms ensures a solid understanding of numbers in any mathematical context. It also provides a foundation to approach more complex problems, where recognizing patterns and converting between formats is necessary for finding solutions.