Square Root of 144/25: Simplified and Explained

To find the square root of the fraction $\frac{144}{25}$, we can separately calculate the square root of the numerator and the denominator.

Step-by-Step Solution

• Find the square root of the numerator: \( \sqrt{144} = 12 \)
• Find the square root of the denominator: \( \sqrt{25} = 5 \)
• Form the fraction with the results: \( \frac{12}{5} \)

Thus, the square root of $\frac{144}{25}$ is $\frac{12}{5}$ or 2.4 in decimal form.

Conclusion

The simplified form of the square root of $\frac{144}{25}$ is $\frac{12}{5}$, which equals 2.4. This method can be used for any fraction by finding the square roots of the numerator and denominator separately.

The square root of a fraction involves finding the square roots of both the numerator and the denominator separately. In this case, we will explore the square root of the fraction $\frac{144}{25}$, a process that is fundamental in mathematics. Understanding this calculation can help in various applications, from simple arithmetic to more complex mathematical problems. Below, we will provide a detailed, step-by-step explanation to make this concept clear and accessible.

First, let’s break down the fraction:

• Numerator: 144
• Denominator: 25

Next, we calculate the square roots of both parts:

1. Find the square root of the numerator: \( \sqrt{144} = 12 \)
2. Find the square root of the denominator: \( \sqrt{25} = 5 \)

Combining these results gives us the simplified form:

\( \sqrt{\frac{144}{25}} = \frac{12}{5} \)

In decimal form, this is equivalent to:

\( \frac{12}{5} = 2.4 \)

Through this straightforward process, we see that the square root of $\frac{144}{25}$ simplifies to \( \frac{12}{5} \), or 2.4. This method can be applied to any fraction, making it a useful tool in various mathematical contexts.

A square root of a number is a value that, when multiplied by itself, gives the original number. For the fraction \( \frac{144}{25} \), we need to find the square root of the numerator and the denominator separately.

First, calculate the square root of 144:

1. Identify that \( 144 = 12 \times 12 \).
2. Therefore, \( \sqrt{144} = 12 \).

Next, calculate the square root of 25:

1. Identify that \( 25 = 5 \times 5 \).
2. Therefore, \( \sqrt{25} = 5 \).

Now, combine these results to find the square root of \( \frac{144}{25} \):

Thus, the square root of \( \frac{144}{25} \) simplifies to \( \frac{12}{5} \).

This process illustrates how to handle square roots in fractions, breaking down each component for clarity and accuracy.

A fraction represents a part of a whole, typically written as \( \frac{a}{b} \) where \( a \) (numerator) is the number of parts considered and \( b \) (denominator) is the total number of equal parts that make up a whole.

In the fraction \( \frac{144}{25} \):

• The numerator \( 144 \) indicates there are 144 parts of something.
• The denominator \( 25 \) indicates the whole is divided into 25 equal parts.

Understanding fractions involves:

1. Identifying the numerator and denominator.
2. Knowing that fractions can represent parts of a whole or division of quantities.
3. Performing operations like addition, subtraction, multiplication, and division on fractions.

For \( \frac{144}{25} \), knowing how to simplify or calculate its square root requires a grasp of basic fraction operations and their application in mathematical contexts.

To calculate \( \sqrt{\frac{144}{25}} \), follow these steps:

1. Find the square root of the numerator \( 144 \):
   • Identify \( 144 = 12 \times 12 \).
   • Thus, \( \sqrt{144} = 12 \).
2. Find the square root of the denominator \( 25 \):
   • Identify \( 25 = 5 \times 5 \).
   • Thus, \( \sqrt{25} = 5 \).
3. Combine the results to simplify the fraction:

   \( \sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5} \).

Therefore, the square root of \( \frac{144}{25} \) is \( \frac{12}{5} \).

To find the square root of 144:

1. Recognize that \( 144 \) is a perfect square, as \( 12 \times 12 = 144 \).
2. Therefore, \( \sqrt{144} = 12 \).

Thus, the square root of 144 is \( 12 \).

To find the square root of 25:

1. Recognize that \( 25 \) is a perfect square, as \( 5 \times 5 = 25 \).
2. Therefore, \( \sqrt{25} = 5 \).

Thus, the square root of 25 is \( 5 \).

To combine the results of \( \sqrt{144} \) and \( \sqrt{25} \) for \( \sqrt{\frac{144}{25}} \):

Therefore, the combined result of \( \sqrt{\frac{144}{25}} \) is \( \frac{12}{5} \).

In mathematics, simplifying fractions involves reducing the fraction to its simplest form where the numerator and denominator are as small as possible and have no common factors other than 1. For the fraction derived from the square root of 144/25, we will follow these steps:

1. Calculate the square roots:

   • √144 = 12
   • √25 = 5

   Thus, √(144/25) = 12/5

2. Check for common factors:

   The fraction 12/5 is already in its simplest form because 12 and 5 have no common factors other than 1.

3. Convert to mixed number (if desired):

   The fraction 12/5 can be expressed as a mixed number:

   • 12 divided by 5 equals 2 with a remainder of 2.
   • Therefore, 12/5 = 2 2/5.

4. Convert to decimal (if desired):

   The fraction 12/5 can be converted to a decimal by dividing the numerator by the denominator:

   • 12 ÷ 5 = 2.4

In summary, the fraction derived from the square root of 144/25 is 12/5, which is already in its simplest form. It can also be expressed as a mixed number, 2 2/5, or as a decimal, 2.4.

To convert the square root of \( \frac{144}{25} \) to decimal form, follow these steps:

1. Start with the expression \( \sqrt{\frac{144}{25}} \). This can be rewritten as the fraction of the square roots of the numerator and the denominator:

   \[ \sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}} \]

2. Next, calculate the square root of the numerator and the denominator separately:

   \[ \sqrt{144} = 12 \]

   \[ \sqrt{25} = 5 \]

3. Now, place these square roots in the fraction:

   \[ \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5} \]

4. Finally, convert the fraction to a decimal by dividing the numerator by the denominator:

   \[ \frac{12}{5} = 2.4 \]

Thus, the decimal form of \( \sqrt{\frac{144}{25}} \) is 2.4.

To find the square root of the fraction \( \frac{144}{25} \), we can simplify this process using the properties of square roots and fractions. The proof involves several steps, which are outlined below:

1. First, rewrite the square root of the fraction as a fraction of square roots:

   \[
   \sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}}
   \]

2. Next, calculate the square roots of the numerator and the denominator separately:

   • For the numerator \(144\):

     \[
     \sqrt{144} = 12
     \]

   • For the denominator \(25\):

     \[
     \sqrt{25} = 5
     \]

3. Now, place the results back into the fraction:

   \[
   \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5}
   \]

4. Therefore, the simplified form of the square root of \( \frac{144}{25} \) is:

   \[
   \sqrt{\frac{144}{25}} = \frac{12}{5}
   \]

5. We can also express the result as a decimal by performing the division:

   \[
   \frac{12}{5} = 2.4
   \]

This proof shows that the square root of \( \frac{144}{25} \) simplifies to \( \frac{12}{5} \), which is equal to 2.4 in decimal form.

Square roots in fractions have numerous practical applications in various fields such as engineering, physics, and finance. Understanding how to work with these can simplify complex calculations and improve problem-solving skills. Here are a few examples:

1. Engineering and Construction: When dealing with geometric shapes and measurements, engineers often use square roots to determine dimensions and areas. For instance, if a material needs to be cut into pieces with an area of \( \frac{144}{25} \) square units, calculating the square root helps find the side length of the square pieces:

   \[
   \text{Side length} = \sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5} = 2.4 \text{ units}
   \]

2. Physics: In physics, square roots in fractions appear frequently when calculating wave functions, energy levels, and other quantities. For example, the square root of the ratio of two energies can help find the ratio of their respective velocities if the energies are proportional to the squares of the velocities:

   \[
   \text{Velocity ratio} = \sqrt{\frac{E_1}{E_2}}
   \]

3. Finance: In finance, square roots are used in risk assessment and portfolio management. The volatility of an investment can be represented as the square root of the variance of returns. If a portfolio's variance is \( \frac{144}{25} \), the volatility (standard deviation) is calculated as:

   \[
   \text{Volatility} = \sqrt{\frac{144}{25}} = 2.4
   \]

4. Statistics: When dealing with normalized data, square roots help in standard deviation calculations and other statistical analyses. If the normalized variance of a dataset is \( \frac{144}{25} \), the standard deviation is:

   \[
   \text{Standard Deviation} = \sqrt{\frac{144}{25}} = 2.4
   \]

These examples demonstrate the versatility and importance of understanding square roots in fractions. Mastering these calculations allows for accurate and efficient problem-solving across various scientific and practical domains.

Working with square roots, especially in fractions like \( \sqrt{\frac{144}{25}} \), can be tricky. Here are some common mistakes to avoid:

1. Incorrectly Simplifying the Fraction: One common error is to incorrectly simplify the fraction inside the square root. Remember that:

   \[
   \sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5}
   \]

   Avoid simplifying \( \frac{144}{25} \) directly without first taking the square roots of the numerator and denominator separately.

2. Forgetting to Simplify Completely: Sometimes, after finding the square roots, the fraction is not simplified further. Always check if the fraction can be reduced:

   \[
   \frac{12}{5} \text{ is already in simplest form.}
   \]

3. Ignoring Negative Roots: While the principal square root is usually positive, remember that every positive number has both a positive and a negative square root. However, in many contexts, only the positive root is considered:

   \[
   \sqrt{144} = 12 \text{ and } -12, \quad \sqrt{25} = 5 \text{ and } -5
   \]

4. Misapplying the Square Root Property: Be cautious not to apply the square root property incorrectly. The property states that:

   \[
   \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
   \]

   Ensure that this is applied correctly to avoid mistakes like \(\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}\).

5. Arithmetic Errors: Simple arithmetic mistakes can lead to incorrect results. Double-check calculations for both the numerator and the denominator:

   \[
   \sqrt{144} = 12 \text{ and } \sqrt{25} = 5 \Rightarrow \frac{12}{5} = 2.4
   \]

By keeping these common mistakes in mind, you can ensure accurate and correct results when dealing with square roots in fractions.

Practicing square root calculations with fractions helps reinforce understanding and accuracy. Below are some practice problems to work through, involving the square root of fractions similar to \( \sqrt{\frac{144}{25}} \).

1. Evaluate the square root of \( \frac{81}{16} \).

   Solution:

   • Simplify the numerator and denominator: \[ \sqrt{\frac{81}{16}} = \frac{\sqrt{81}}{\sqrt{16}} = \frac{9}{4} \]
   • Convert to decimal form: \[ \frac{9}{4} = 2.25 \]

2. Find the square root of \( \frac{49}{36} \).

   Solution:

   • Simplify the fraction under the square root: \[ \sqrt{\frac{49}{36}} = \frac{\sqrt{49}}{\sqrt{36}} = \frac{7}{6} \]
   • Convert to decimal form: \[ \frac{7}{6} \approx 1.1667 \]

3. Simplify \( \sqrt{\frac{64}{25}} \).

   Solution:

   • Simplify the fraction under the square root: \[ \sqrt{\frac{64}{25}} = \frac{\sqrt{64}}{\sqrt{25}} = \frac{8}{5} \]
   • Convert to decimal form: \[ \frac{8}{5} = 1.6 \]

4. Evaluate \( \sqrt{\frac{100}{49}} \).

   Solution:

   • Simplify the fraction under the square root: \[ \sqrt{\frac{100}{49}} = \frac{\sqrt{100}}{\sqrt{49}} = \frac{10}{7} \]
   • Convert to decimal form: \[ \frac{10}{7} \approx 1.4286 \]

5. Simplify and evaluate \( \sqrt{\frac{121}{81}} \).

   Solution:

   • Simplify the fraction under the square root: \[ \sqrt{\frac{121}{81}} = \frac{\sqrt{121}}{\sqrt{81}} = \frac{11}{9} \]
   • Convert to decimal form: \[ \frac{11}{9} \approx 1.2222 \]

These practice problems cover various fractions to help you gain confidence in working with square roots. Remember to always simplify the numerator and denominator separately before combining the results.

To further enhance your understanding of square roots and their applications, especially in the context of fractions like 144/25, the following resources are recommended:

• This resource provides a comprehensive guide on how to simplify square root expressions, including those involving fractions.

• An excellent introductory resource that covers the basics of square roots, including properties and simplification techniques.

• This site offers a detailed explanation of square roots, visual aids, and interactive exercises to practice what you've learned.

• Focused on the specific topic of simplifying the square roots of fractions, this resource includes step-by-step examples.

• A thorough guide on finding and simplifying the square root of a fraction, with additional practice problems and solutions.

For those looking to deepen their knowledge through textbooks and other comprehensive materials, consider the following:

1. “Algebra and Trigonometry” by Michael Sullivan:

   This textbook provides an in-depth look at algebraic principles, including a dedicated section on radicals and their applications.

2. “Precalculus: Mathematics for Calculus” by James Stewart, Lothar Redlin, and Saleem Watson:

   A valuable resource for students preparing for calculus, this book covers essential topics such as square roots and fractional expressions.

Additionally, you can explore online courses and video tutorials that offer interactive learning experiences:

• This series of courses covers fundamental algebraic concepts, including radicals and fractions, with interactive assignments and peer-reviewed projects.

• An accessible course for beginners that breaks down complex algebraic topics into understandable lessons, including square roots and their applications.