Problem 7 - Intermediate Algebra

Question

Tell why each of the following expressions is not in simplified radical form. Then simplify it. Finally, use a calculator to approximate its value. $$ \begin{array}{|l|l|l|l|} \hline & \begin{array}{l} \text { Why isn't it in } \\ \text { simplified form? } \end{array} & \begin{array}{l} \text { Simplified } \\ \text { form } \end{array} & \text { Approximation } \\ \hline \frac{3}{\sqrt{2}} & & \\ \hline \frac{\sqrt{18}}{2} & & \\ \hline \sqrt{\frac{9}{2}} & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Each expression is not simplified due to radicals in denominators or non-prime radicals. Simplified forms yield \( \frac{3\sqrt{2}}{2} \) for all three expressions, approximating to 2.1213 with a calculator.

Step 1: Analyze Expression 1

The expression \( \frac{3}{\sqrt{2}} \) is not in simplified form because it has a radical in the denominator. Rationalizing the denominator can simplify it.

Step 2: Simplify Expression 1

Multiply the numerator and the denominator by \( \sqrt{2} \) to remove the square root from the denominator: \[ \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \]. The simplified form is \( \frac{3\sqrt{2}}{2} \).

Step 3: Approximate Expression 1

Calculate \( \frac{3\sqrt{2}}{2} \) using a calculator: \( \sqrt{2} \approx 1.4142 \) then \( 3 \times 1.4142 \div 2 \approx 2.1213 \).

Step 4: Analyze Expression 2

The expression \( \frac{\sqrt{18}}{2} \) is not simplified because \( \sqrt{18} \) can be simplified further. It can be broken down into a product of simpler square roots.

Step 5: Simplify Expression 2

Simplify \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \). Thus, \[ \frac{\sqrt{18}}{2} = \frac{3\sqrt{2}}{2} \].

Step 6: Approximate Expression 2

The approximation will be the same as for Expression 1 since the simplified form is identical: \( \frac{3\sqrt{2}}{2} \approx 2.1213 \).

Step 7: Analyze Expression 3

The expression \( \sqrt{\frac{9}{2}} \) is not in simplified form because the fraction inside the square root can be simplified to separate the numerator and the denominator.

Step 8: Simplify Expression 3

Apply the square root property: \( \sqrt{\frac{9}{2}} = \frac{\sqrt{9}}{\sqrt{2}} = \frac{3}{\sqrt{2}} \), then rationalize the denominator: \[ \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \].

Step 9: Approximate Expression 3

Use a calculator to approximate the value as previously calculated: \( \frac{3\sqrt{2}}{2} \approx 2.1213 \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a vital step in simplifying radical expressions. When we see a square root in the denominator, like in the expression \( \frac{3}{\sqrt{2}} \), it is not considered to be fully simplified. This is because roots in the denominator can complicate division calculations and are typically non-standard in mathematical practice.

To rationalize the denominator, we multiply the numerator and the denominator by the same square root, essentially using the identity property (multiplying by one) to remove the root.

For example:

• Given: \( \frac{3}{\sqrt{2}} \).
• Multiply by \( \frac{\sqrt{2}}{\sqrt{2}} \): \( \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \).

Now, the expression is \( \frac{3\sqrt{2}}{2} \) with no square root in the denominator, making it rationalized and simplified.

Square Roots

Square roots serve as operations that identify a number which, when multiplied by itself, gives the original number. Understanding square roots is crucial when working with expressions like \( \sqrt{18} \).

Breaking down a square root can help in obtaining a simpler form:

• \( \sqrt{18} = \sqrt{9 \times 2} \).
• Now separate: \( \sqrt{9} \times \sqrt{2} \).
• This simplifies to \( 3\sqrt{2} \) since \( \sqrt{9} = 3 \).

This simplification is frequently needed to find a more streamlined representation of expressions and to make further calculations more manageable. It helps identify patterns and relations in equations, leading to a deeper understanding of the processes involved.

Approximating Values

Approximating values is the process of finding a numerical value close to the exact solution, providing easy comparison or practical usage. It is especially useful when dealing with irrational numbers—such as roots—that do not result in a simple fractional equivalent.

For example, when approximating the value of \( \frac{3\sqrt{2}}{2} \):

• First, approximate \( \sqrt{2} \approx 1.4142 \).
• Then calculate: \( 3 \times 1.4142 \div 2 \).
• This results in approximately \( 2.1213 \).

Using a calculator, students can quickly and efficiently find such approximations, enabling them to relate more abstract mathematical expressions to real-world numbers, which aids in better understanding and application.