What is the Square Root of 108? Discover the Answer and Its Applications

The square root of 108 can be represented in several ways. Below are the different representations and methods to find the square root of 108.

Decimal Form

The square root of 108 in decimal form is approximately:

\(\sqrt{108} \approx 10.3923\)

Simplified Radical Form

The square root of 108 can also be simplified to its radical form:

\(\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}\)

Prime Factorization Method

Using prime factorization, we can break down 108 as follows:

• 108 = 2 × 2 × 3 × 3 × 3
• Group the prime factors into pairs: (2 × 2) and (3 × 3) with a single 3 remaining

Thus, the square root of 108 is:

\(\sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}\)

Using a Calculator

You can also use a calculator to find the square root of 108, which will give you the decimal approximation:

\(\sqrt{108} \approx 10.3923\)

Visual Representation

To visualize the square root of 108, consider a square with an area of 108 square units. The length of each side of this square will be the square root of 108, approximately 10.3923 units.

The square root of 108 is a number that, when multiplied by itself, equals 108. It is an important concept in mathematics, particularly in algebra and geometry. Understanding the square root of 108 involves breaking it down into simpler components and exploring its decimal and radical forms.

Here are the key points to understand about the square root of 108:

• Decimal Representation: The square root of 108 in decimal form is approximately 10.3923.
• Radical Form: The square root of 108 can be expressed as \(6\sqrt{3}\) in its simplest radical form.
• Prime Factorization: Breaking down 108 into prime factors helps in simplifying the square root:
  1. 108 = 2 × 2 × 3 × 3 × 3
  2. Grouping the prime factors: (2 × 2) and (3 × 3) with a single 3 remaining
  3. Simplifying gives: \(\sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}\)
• Using a Calculator: For quick calculations, you can use a calculator to find the square root of 108, which provides a decimal approximation of 10.3923.

Understanding the square root of 108 is beneficial for solving mathematical problems and has practical applications in various fields such as physics, engineering, and computer science. This guide will delve deeper into each aspect, ensuring a comprehensive understanding of the square root of 108.

The decimal representation of the square root of 108 provides a precise value that can be useful in various calculations and applications. To find the decimal form, we typically use a calculator or a computational method. Here's a detailed look at the process:

The square root of 108 is approximately:

\(\sqrt{108} \approx 10.392304845413264\)

This value is often rounded to a more manageable number of decimal places for practical purposes. Commonly, it is rounded to four or five decimal places:

• Rounded to four decimal places: 10.3923
• Rounded to five decimal places: 10.39230

Understanding the decimal representation can be particularly useful in the following scenarios:

• Precision Calculations: When exact measurements are required, using the full decimal value ensures accuracy.
• Engineering Applications: Engineers often use decimal representations for precise measurements and calculations.
• Scientific Research: Scientists rely on decimal values for detailed data analysis and experiments.

For further accuracy, advanced calculators or computer software can provide even more decimal places. However, for most everyday purposes, rounding to four or five decimal places is sufficient.

To summarize, the square root of 108 in its decimal form is a key numerical value that serves various practical purposes in science, engineering, and everyday calculations.

The simplified radical form of the square root of 108 is a more compact and exact way to express this value, using square roots and integers. This method is particularly useful in algebra and higher mathematics. Here’s how to simplify the square root of 108 step by step:

1. Prime Factorization:

   First, break down 108 into its prime factors:

   108 = 2 × 2 × 3 × 3 × 3

2. Group the Factors:

   Next, group the prime factors into pairs of the same number:

   (2 × 2) and (3 × 3) with one 3 remaining unpaired.

3. Extract the Square Roots:

   Take the square root of each pair of factors:

   \(\sqrt{2^2} = 2\) and \(\sqrt{3^2} = 3\)

4. Multiply the Results:

   Combine the extracted square roots and the remaining factor under the radical:

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

Thus, the simplified radical form of the square root of 108 is:

\(\sqrt{108} = 6\sqrt{3}\)

This form is useful in mathematical proofs and equations where exact values are necessary. Here’s a summary of the key points:

• \(\sqrt{108}\) can be expressed as \(6\sqrt{3}\).
• Prime factorization helps in breaking down the number into simpler components.
• Extracting square roots from pairs of factors simplifies the expression.

Understanding the simplified radical form of the square root of 108 not only helps in mathematical computations but also in grasping the underlying structure of numbers and their properties.

Prime factorization is a method of breaking down a number into its prime number factors. To find the square root of 108 using prime factorization, follow these steps:

1. First, find the prime factors of 108.
• 108 is an even number, so start by dividing it by 2:

\[
108 \div 2 = 54
\]

• 54 is also even, so divide by 2 again:

\[
54 \div 2 = 27
\]

• 27 is divisible by 3:

\[
27 \div 3 = 9
\]

• 9 is also divisible by 3:

\[
9 \div 3 = 3
\]

• Finally, 3 is a prime number, so we stop here.

So, the prime factors of 108 are:

\[
108 = 2 \times 2 \times 3 \times 3 \times 3
\]

2. Group the prime factors into pairs.

\[
108 = (2 \times 2) \times (3 \times 3) \times 3
\]

3. Take the square root of each pair and multiply them.

\[
\sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} = \sqrt{2^2 \times 3^2 \times 3}
\]

Using the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\):

\[
\sqrt{2^2 \times 3^2 \times 3} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{3}
\]

4. Simplify the square roots of the pairs.

\[
\sqrt{2^2} = 2
\]

\[
\sqrt{3^2} = 3
\]

So:

\[
\sqrt{2^2 \times 3^2 \times 3} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}
\]

Therefore, the square root of 108 in its simplest radical form is \(6\sqrt{3}\).

Finding the square root of 108 using a calculator is straightforward. Here's a detailed step-by-step guide on how to do it:

1. Turn on your calculator.
2. Enter the number 108.
3. Press the square root button (√) on your calculator. This button is typically marked with the symbol √ or labeled as "sqrt".
4. The calculator will display the square root of 108. The value should be approximately 10.392304845413.

Here are the steps demonstrated using a scientific calculator:

• Turn on the calculator.
• Input 108 using the number keys.
• Press the √ key. The result shown will be approximately 10.392304845413.

If you're using an online calculator, follow these instructions:

1. Open your web browser and navigate to an online square root calculator, such as or .
2. Enter 108 in the input field.
3. Click the calculate button.
4. The square root will be displayed on the screen, which is 10.392304845413.

You can also use spreadsheet software like Microsoft Excel or Google Sheets to find the square root:

• Open Excel or Google Sheets.
• Select a cell and type =SQRT(108).
• Press Enter, and the cell will display the square root of 108, which is 10.392304845413.

Using a calculator to find the square root of 108 is quick and accurate, providing you with an answer in both decimal and simplified forms as needed.

Visualizing the square root of 108 can help in understanding its value and applications. Below are various ways to represent the square root of 108 visually:

Number Line Representation

A number line is a straightforward way to visualize the square root of 108:

• Identify the perfect squares around 108, which are 100 (10²) and 121 (11²).
• √108 lies between 10 and 11 on the number line, closer to 10.4 since √108 ≈ 10.392.

Here is a graphical representation:

Geometric Representation

We can also represent √108 geometrically as the length of the diagonal of a square or rectangle:

• Consider a rectangle with sides of length √54 and √2 (since 54 * 2 = 108).
• The diagonal of such a rectangle will be √108.

Using the Pythagorean theorem, the diagonal (d) of a rectangle with sides a and b is given by:

\[
d = \sqrt{a^2 + b^2}
\]

For a square with side length \(a = \sqrt{54}\):

\[
d = \sqrt{(\sqrt{54})^2 + (\sqrt{2})^2} = \sqrt{54 + 2} = \sqrt{108}
\]

Graphical Calculator Representation

Using a graphing calculator or software like Desmos, you can plot the function \(y = \sqrt{x}\) and find the point where \(x = 108\). This will visually show the square root of 108 on a graph.

These visual representations help in comprehending the magnitude and significance of the square root of 108.

The square root of 108, approximately 10.39, can be applied in various real-life scenarios across different fields. Here are some practical examples:

• Geometry and Construction:

  In geometry, the square root is used to calculate distances and measurements. For example, if you are designing a triangular structure and need to find the length of a side when the lengths of the other sides are known, you can apply the Pythagorean theorem, which involves square roots.

  The formula used is \( c = \sqrt{a^2 + b^2} \). In construction, accurate measurements are crucial for ensuring stability and safety.

• Physics and Engineering:

  Square roots are often used in physics to determine quantities like velocity and acceleration. For example, the formula to calculate the root mean square (RMS) speed of particles in a gas is \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a particle.

  In electrical engineering, square roots are used to calculate the effective values of alternating current (AC) and voltage.

• Finance and Economics:

  Square roots are used in finance to calculate the standard deviation and variance of investment returns. These metrics are essential for assessing the risk and volatility of investments. The formula for standard deviation \( \sigma \) is \( \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2} \), where \( \mu \) is the mean return and \( N \) is the number of observations.

• Statistics:

  In statistics, the square root is used to compute the standard deviation and variance, which are key measures of data dispersion. For example, if you have a set of data points and need to understand how spread out they are, you calculate the variance first and then take the square root to get the standard deviation.

• Navigation and Mapping:

  Square roots are used in navigation to calculate the shortest distance between two points on a map. The distance formula \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) helps in plotting and finding accurate paths in 2D space, essential for fields like aviation and maritime navigation.

These applications show the versatility and importance of square roots in solving real-world problems efficiently and accurately.

Here are some frequently asked questions about the square root of 108:

• What is the square root of 108?

The square root of 108 is approximately 10.3923048454. It can also be expressed in simplest radical form as \( 6\sqrt{3} \).

• Is the square root of 108 a rational number?

No, the square root of 108 is an irrational number. This is because its decimal representation is non-terminating and non-repeating.

• How can the square root of 108 be simplified?

The square root of 108 can be simplified using prime factorization. The prime factors of 108 are 2, 3, and 3. Therefore, \( \sqrt{108} = \sqrt{2^2 \times 3^3} = 6\sqrt{3} \).

• How is the square root of 108 calculated using a calculator?

To find the square root of 108 using a calculator, simply enter "108" and press the square root (√) button. The result should be approximately 10.3923048454.

• Can the square root of 108 be negative?

Mathematically, the square root of 108 can have both a positive and negative value, represented as \( \pm10.3923048454 \). However, in most contexts, the positive root is used.

• What are some applications of the square root of 108?

The square root of 108 can be used in various mathematical contexts, such as geometry (e.g., calculating distances or areas), algebra, and real-life applications like physics and engineering where root values are necessary for computations.

To deepen your understanding of square roots, there are numerous educational resources available online. Here are some highly recommended sources:

• This site provides a comprehensive overview of square roots, including explanations, examples, and interactive elements to help visualize the concept. It's an excellent resource for beginners and those looking to reinforce their understanding.

• Khan Academy offers detailed video tutorials and practice exercises on square roots. The videos cover basic to advanced topics, making it suitable for a wide range of learners. The interactive exercises help reinforce the concepts learned in the videos.

• This YouTube video by Khan Academy provides an engaging introduction to square roots, explaining the fundamental concepts in an easy-to-understand manner. It is ideal for visual learners who benefit from step-by-step explanations.

• Sofatutor offers a series of videos and interactive lessons that simplify the concept of square roots. The platform's approach is designed to make learning fun and accessible, with plenty of examples and practice problems.

• This website provides in-depth lessons on square roots, including how to simplify square root expressions and understand their properties. The lessons are well-structured and cater to both beginners and advanced students.

These resources will provide you with a thorough understanding of square roots and help you apply these concepts in various mathematical problems.