6 Divided by Square Root of 2: Simplified Calculation Guide

To simplify the expression \( \frac{6}{\sqrt{2}} \), follow these steps:

Step-by-Step Solution

1. Start with the expression: \[ \frac{6}{\sqrt{2}} \]
2. Rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{2} \): \[ \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6 \sqrt{2}}{2} \]
3. Simplify the fraction: \[ \frac{6 \sqrt{2}}{2} = 3 \sqrt{2} \]

Thus, the simplified form of \( \frac{6}{\sqrt{2}} \) is \( 3 \sqrt{2} \).

Decimal Form

To express \( 3 \sqrt{2} \) in decimal form, calculate:

Therefore, \( 3 \sqrt{2} \approx 4.242 \).

Further Explanation and Practice

The process of rationalizing the denominator involves removing the square root from the denominator by multiplying both the numerator and the denominator by the square root found in the denominator. This method ensures that the expression is in its simplest form.

Example Problems

• Simplify \( \frac{8}{\sqrt{3}} \): \[ \frac{8}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{8 \sqrt{3}}{3} \]
• Simplify \( \frac{5}{\sqrt{7}} \): \[ \frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{5 \sqrt{7}}{7} \]

The calculation of dividing 6 by the square root of 2, denoted as \( \frac{6}{\sqrt{2}} \), is a common problem in mathematics. This operation involves simplifying the expression by rationalizing the denominator. The process transforms the initial expression into a more manageable form without a radical in the denominator, making it easier to handle in further calculations.

To simplify \( \frac{6}{\sqrt{2}} \), follow these steps:

1. Multiply both the numerator and the denominator by \( \sqrt{2} \).
2. This gives \( \frac{6 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \).
3. Simplify the denominator \( \sqrt{2} \cdot \sqrt{2} \) to 2.
4. The expression becomes \( \frac{6\sqrt{2}}{2} \).
5. Finally, divide 6 by 2 to get \( 3\sqrt{2} \).

Therefore, \( \frac{6}{\sqrt{2}} \) simplifies to \( 3\sqrt{2} \). This simplification is useful in various mathematical contexts, ensuring clarity and precision in calculations involving radicals.

Dividing 6 by the square root of 2, represented as \( \frac{6}{\sqrt{2}} \), involves a process known as rationalizing the denominator. The goal is to eliminate the radical in the denominator for a simpler, more usable form. Here's a detailed step-by-step explanation of the process:

1. Identify the initial expression: \( \frac{6}{\sqrt{2}} \).
2. Multiply both the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator:

   
   \( \frac{6 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \).

3. Simplify the denominator:

   
   Since \( \sqrt{2} \cdot \sqrt{2} = 2 \), the expression becomes \( \frac{6\sqrt{2}}{2} \).

4. Divide the numerator by the simplified denominator:

   
   \( \frac{6\sqrt{2}}{2} = 3\sqrt{2} \).

Thus, \( \frac{6}{\sqrt{2}} \) simplifies to \( 3\sqrt{2} \). This simplified form is often more convenient for further mathematical operations and applications.

To simplify the expression \( \frac{6}{\sqrt{2}} \), follow these steps:

1. Start with the original expression: \[ \frac{6}{\sqrt{2}} \]
2. Rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{2} \): \[ \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2} \]
3. Simplify the fraction by dividing the numerator by the denominator: \[ \frac{6\sqrt{2}}{2} = 3\sqrt{2} \]

The simplified form of \( \frac{6}{\sqrt{2}} \) is \( 3\sqrt{2} \).

Understanding how to divide numbers involving square roots can be further reinforced through practical examples and practice problems. Here are a few examples and problems for you to try:

Example 1: Simplifying \(\frac{6}{\sqrt{2}}\)

Step-by-step solution:

1. Rewrite the expression: \(\frac{6}{\sqrt{2}}\).
2. Rationalize the denominator by multiplying the numerator and the denominator by \(\sqrt{2}\):

\(\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}\).

Example 2: Simplifying \(\frac{10}{\sqrt{5}}\)

Step-by-step solution:

1. Rewrite the expression: \(\frac{10}{\sqrt{5}}\).
2. Rationalize the denominator by multiplying the numerator and the denominator by \(\sqrt{5}\):

\(\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}\).

Practice Problems

Test your knowledge with these practice problems:

• Simplify \(\frac{8}{\sqrt{3}}\).
• Simplify \(\frac{12}{\sqrt{6}}\).
• Simplify \(\frac{5}{\sqrt{2}}\).
• Simplify \(\frac{9}{\sqrt{7}}\).

Solutions:

• \(\frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3}\)
• \(\frac{12}{\sqrt{6}} = \frac{12\sqrt{6}}{6} = 2\sqrt{6}\)
• \(\frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2}\)
• \(\frac{9}{\sqrt{7}} = \frac{9\sqrt{7}}{7}\)

Utilizing tools and calculators can greatly simplify the process of working with complex mathematical expressions, such as dividing 6 by the square root of 2. Here are some recommended resources:

• Scientific Calculators: High-quality scientific calculators, like those from Texas Instruments or Casio, have built-in functions to handle square roots and other complex operations. They are essential tools for students and professionals.
• Online Math Solvers: Websites such as Wolfram Alpha and Symbolab offer step-by-step solutions for various mathematical problems. You can input equations and get detailed explanations on how to solve them.
• Mobile Apps: There are numerous mobile applications available that can perform complex calculations on the go. These apps often include features for graphing and algebraic simplifications, making them convenient for quick calculations.

These tools are invaluable for enhancing understanding and efficiency when working on mathematical problems, whether for academic purposes or practical applications.

When dividing numbers by the square root of 2, several common mistakes can occur. Here are some tips to avoid these errors and ensure accuracy in your calculations.

• Not Rationalizing the Denominator: Always remember to rationalize the denominator. For instance, when dividing 6 by the square root of 2, multiply both the numerator and the denominator by the square root of 2 to simplify.
• Incorrect Simplification: Ensure that you correctly simplify both the numerator and the denominator. Simplification errors can lead to incorrect results.
• Forgetting to Multiply Both Terms: When rationalizing, don't forget to multiply both the numerator and the denominator by the same term to maintain the equality of the fraction.
• Ignoring Square Root Properties: Remember that \(\sqrt{a} \times \sqrt{a} = a\). Use this property to help simplify your calculations correctly.

Here are the step-by-step tips to avoid these common mistakes:

1. Step 1: Identify the need to rationalize the denominator. For example, if you have \(\frac{6}{\sqrt{2}}\), you need to eliminate the square root from the denominator.
2. Step 2: Multiply both the numerator and the denominator by the square root of 2: \[ \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2} \]
3. Step 3: Simplify the resulting fraction. Here, \(\frac{6\sqrt{2}}{2} = 3\sqrt{2}\).
4. Step 4: Verify your final answer to ensure no errors were made during simplification.

By following these steps and keeping these tips in mind, you can avoid common mistakes and perform your calculations correctly and efficiently.

Further reading and resources for in-depth understanding: