Square Root of 1/144: Simplified Explanation and Calculation

The square root of a fraction can be found by taking the square root of the numerator and the denominator separately. Here, we find the square root of 1/144.

Step-by-Step Solution

1. Identify the fraction: \(\frac{1}{144}\)
2. Take the square root of the numerator: \(\sqrt{1} = 1\)
3. Take the square root of the denominator: \(\sqrt{144} = 12\)
4. Combine the results: \(\frac{\sqrt{1}}{\sqrt{144}} = \frac{1}{12}\)

Result

The square root of \(\frac{1}{144}\) is \(\frac{1}{12}\).

Mathematical Representation

Using Mathjax, the equation can be represented as:

\[\sqrt{\frac{1}{144}} = \frac{\sqrt{1}}{\sqrt{144}} = \frac{1}{12}\]

Conclusion

The calculation shows that the square root of \(\frac{1}{144}\) is \(\frac{1}{12}\), confirming the method used is correct and reliable.

Square roots are fundamental in mathematics, representing a value that, when multiplied by itself, gives the original number. Understanding square roots is crucial for solving various mathematical problems and equations.

The square root of a number \(x\) is written as \(\sqrt{x}\). For example, the square root of 9 is 3, because \(3 \times 3 = 9\).

Square roots can be applied to both whole numbers and fractions. When dealing with fractions, the square root of a fraction can be found by taking the square root of the numerator and the denominator separately.

Let's break down the process:

1. Identify the fraction: For instance, \(\frac{1}{144}\).
2. Take the square root of the numerator: \(\sqrt{1} = 1\).
3. Take the square root of the denominator: \(\sqrt{144} = 12\).
4. Combine the results: \(\frac{\sqrt{1}}{\sqrt{144}} = \frac{1}{12}\).

Therefore, the square root of \(\frac{1}{144}\) is \(\frac{1}{12}\).

Understanding square roots and their applications can significantly enhance your mathematical proficiency and problem-solving abilities.

The fraction $$

1
144

represents a division of 1 by 144. To fully grasp this fraction, let's break down its components and understand their significance.

Numerator: The numerator is the top number in a fraction. In this case, it is 1. The numerator indicates how many parts we have.

Denominator: The denominator is the bottom number in a fraction. Here, it is 144. The denominator shows into how many equal parts the whole is divided.

Therefore, the fraction $$

1
144

means we are looking at one part out of 144 equal parts.

Properties of 1/144

• Proper Fraction: Since the numerator (1) is less than the denominator (144), it is a proper fraction.
• Value: The decimal equivalent of $\frac{1}{144}$ is approximately 0.00694.
• Simplification: The fraction $\frac{1}{144}$ is already in its simplest form because 1 and 144 have no common factors other than 1.

Understanding these basics about the fraction $$

1
144

will aid in comprehending its square root and other mathematical operations involving this fraction.

To fully understand the process of finding the square root of the fraction \( \frac{1}{144} \), we need to break down both the numerator and the denominator. This will help us understand how the square root applies to each part of the fraction separately.

The fraction \( \frac{1}{144} \) consists of two parts:

• The numerator: 1
• The denominator: 144

Let's analyze each component:

Numerator

The numerator is 1. The square root of 1 is straightforward because:

\[ \sqrt{1} = 1 \]

This is because 1 multiplied by itself always equals 1.

Denominator

The denominator is 144. To find the square root of 144, we consider its factors:

144 is a perfect square, which means it can be expressed as:

\[ 144 = 12 \times 12 \]

Thus, the square root of 144 is:

\[ \sqrt{144} = 12 \]

Combining the Results

Now, we combine the square roots of the numerator and the denominator:

\[ \sqrt{\frac{1}{144}} = \frac{\sqrt{1}}{\sqrt{144}} = \frac{1}{12} \]

Therefore, the square root of \( \frac{1}{144} \) is \( \frac{1}{12} \).

To calculate the square root of the numerator \(1\) in \( \frac{1}{144} \), we proceed as follows:

1. Recognize that the numerator is \(1\).
2. Apply the square root operation: \( \sqrt{1} = 1 \).

Therefore, the square root of the numerator \(1\) is \(1\).

To calculate the square root of the denominator \(144\) in \( \frac{1}{144} \), follow these steps:

1. Identify the denominator as \(144\).
2. Apply the square root operation: \( \sqrt{144} = 12 \).

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

Now, let's combine the results of calculating the square roots of the numerator and the denominator in \( \frac{1}{144} \):

Therefore, the square root of \( \frac{1}{144} \) is:

To verify that \( \sqrt{\frac{1}{144}} = \frac{1}{12} \), follow these steps:

1. Calculate \( \left(\frac{1}{12}\right)^2 \):

\( \left(\frac{1}{12}\right)^2 \)

\( = \frac{1}{144} \)

2. Since \( \left(\frac{1}{12}\right)^2 = \frac{1}{144} \), this confirms that \( \sqrt{\frac{1}{144}} = \frac{1}{12} \).

Therefore, the result \( \sqrt{\frac{1}{144}} = \frac{1}{12} \) is verified.

Square roots are fundamental in various mathematical applications:

1. Geometry: They are used to calculate side lengths of squares and other geometric figures.
2. Physics: Square roots appear in equations involving energy, velocity, and acceleration.
3. Statistics: They are essential in calculating standard deviation and other measures of dispersion.
4. Engineering: Square roots are used in solving problems related to electrical circuits and signal processing.
5. Finance: They play a role in calculating interest rates and investment returns.
6. Computer Science: Algorithms involving square roots are used in various computational tasks.

Understanding square roots is crucial for both theoretical understanding and practical applications across different fields of mathematics and beyond.

When dealing with the square root of \( \frac{1}{144} \), it's important to avoid the following mistakes:

• Forgetting to simplify the square root expression properly.
• Misinterpreting the result of \( \sqrt{\frac{1}{144}} \) as \( \frac{\sqrt{1}}{\sqrt{144}} \), which is incorrect.
• Confusing the calculation by not understanding that \( \sqrt{\frac{1}{144}} \) simplifies directly to \( \frac{1}{12} \).
• Incorrectly applying the square root operation to only the numerator or denominator of the fraction.
• Overlooking the verification step to confirm that \( \sqrt{\frac{1}{144}} = \frac{1}{12} \).

By avoiding these common mistakes, one can ensure accurate calculation and understanding of the square root of \( \frac{1}{144} \).

Here are some practice problems involving the square root of \( \frac{1}{144} \):

1. Calculate \( \sqrt{\frac{1}{144}} \).
2. Verify your result by squaring \( \frac{1}{12} \) and confirming it equals \( \frac{1}{144} \).
3. Solve for \( x \) in the equation \( \sqrt{x} = \frac{1}{12} \).
4. Find the square root of \( \frac{4}{144} \).
5. Determine whether \( \sqrt{\frac{1}{169}} \) is greater than or less than \( \frac{1}{12} \).

These problems will help reinforce your understanding and calculation skills related to square roots.

For further exploration into square roots and related topics, consider the following resources:

• Advanced textbooks on algebra and calculus that delve into square root properties.
• Online courses or tutorials that provide interactive lessons on mathematical concepts.
• Mathematical forums and communities where discussions on square roots and fractional calculations are frequent.
• Additional practice problems and exercises to strengthen your skills in calculating square roots.
• Research papers that explore applications of square roots in various scientific disciplines.

Continuing to study and practice will enhance your understanding and proficiency in dealing with square roots and their applications.