Square Root of 117 Simplified: Discover the Simplest Radical Form

The square root of 117 can be expressed in its simplest radical form as:

$$\sqrt{117} = 3\sqrt{13}$$

Here is a detailed explanation on how to achieve this simplification:

Step-by-Step Simplification Process

1. Factor 117 into its prime factors:
   • 117 = 3 × 3 × 13
2. Group the prime factors:
   • 117 = \(3^2 \times 13\)
3. Apply the square root to each group:
   • $$\sqrt{117} = \sqrt{3^2 \times 13} = \sqrt{3^2} \times \sqrt{13} = 3\sqrt{13}$$

Therefore, the simplified form of the square root of 117 is 3√13.

Additional Information

The decimal approximation of √117 is approximately 10.8167. This value is irrational, meaning it cannot be expressed as a simple fraction.

Related Examples

• Square root of 118: 10.8628
• Square root of 119: 10.9087
• Square root of 120: 10.9545

For more detailed steps and examples, you can refer to educational resources like Mathway, Cuemath, and MathOnDemand.

The square root of 117 can be simplified to help understand its mathematical properties better. Simplifying the square root involves expressing it in its simplest radical form, which is particularly useful in various mathematical contexts. Here, we will explore the steps to simplify the square root of 117 and discuss its significance.

First, recognize that 117 is not a perfect square, meaning its square root will not be an integer. We use the prime factorization method to simplify the square root:

1. Find the prime factors of 117, which are 3, 3, and 13. Thus, 117 can be expressed as 3² * 13.
2. Rewrite the square root of 117 as the product of the square roots of its prime factors: √117 = √(3² * 13).
3. Since the square root of a product is the product of the square roots, we get: √(3² * 13) = √(3²) * √13.
4. The square root of 3² is 3, so we simplify this to: 3√13.

Therefore, the simplified form of the square root of 117 is 3√13, which approximately equals 10.816653826392 in decimal form.

This simplification is useful for mathematical problem-solving and provides a clearer understanding of the properties of the number 117.

To simplify the square root of 117 using the prime factorization method, follow these detailed steps:

1. First, find the prime factors of 117. Start by dividing 117 by the smallest prime number, which is 2. Since 117 is not divisible by 2, move to the next prime number, 3.
2. Divide 117 by 3:

   117 ÷ 3 = 39

   39 is also divisible by 3:
   
   39 ÷ 3 = 13

   13 is a prime number.

3. Thus, the prime factorization of 117 is:
   117 = 3 × 3 × 13
4. Group the prime factors into pairs:

   (3 × 3) and 13

5. Take one factor from each pair and multiply them:

   √117 = √(3² × 13) = 3√13

Therefore, the simplified form of the square root of 117 is 3√13.

In decimal form, this is approximately 10.816653826392.

The square root of 117 can be expressed in decimal form, which is useful for various practical applications. To find the decimal form of the square root of 117, you can use a calculator or perform long division.

The value of the square root of 117 in decimal form is approximately 10.816653826392.

Step-by-Step Calculation

Here's how you can calculate it step-by-step:

1. Identify the perfect squares closest to 117. In this case, 100 and 121 are the perfect squares nearest to 117. This indicates that the square root of 117 lies between 10 and 11.
2. To find a more precise value, you can use a calculator or the long division method.

Using a Calculator

Using a calculator, simply enter 117 and press the square root (√) button. The result will be approximately 10.8167.

Long Division Method

The long division method provides a more detailed approach:

1. Pair the digits of 117 from right to left (1 17.00).
2. Find the largest number whose square is less than or equal to the first pair (1). In this case, it is 1. Write 1 above the line and subtract 1 from the first pair, resulting in 0. Bring down the next pair of digits (17).
3. Double the number above the line (1), which gives 2. Determine how many times 2 can be multiplied to get a product close to or less than 17. The answer is 8 (20 * 8 = 160). Write 8 above the line next to 1 and subtract the product from 1700, bringing down the next pair of zeros.
4. Continue the process to obtain the decimal places. This will eventually give you 10.816.

Rounded Values

For practical purposes, the square root of 117 can be rounded to different decimal places:

• Nearest tenth: 10.8
• Nearest hundredth: 10.82
• Nearest thousandth: 10.817

Understanding the decimal form of the square root of 117 is essential for various mathematical and real-world applications, providing a precise and useful representation of this irrational number.

The square root of 117 has several interesting properties. Here, we explore its characteristics and how it compares to other square roots.

• Perfect Square: 117 is not a perfect square. This means its square root will not be an integer.
• Irrational Number: The square root of 117 is an irrational number. This is because it cannot be expressed as a simple fraction.
• Comparison with Nearest Perfect Squares:
  • The closest perfect squares to 117 are 100 (10²) and 121 (11²).
  • This places √117 between 10 and 11.
• Exponential Form: The square root of a number can be expressed in exponential form. For example, √117 can be written as 117^1/2.
• Even/Odd Nature:
  • The square root of an even number is always even. For example, √64 = 8.
  • The square root of an odd number is always odd. Since 117 is odd, √117 will be an odd number (irrational in this case).
• Significance of Last Digit:
  • If a number ends in 2, 3, 7, or 8, its square root is not an integer. Since 117 ends in 7, its square root is not an integer.
• Multiplication Rule: The square root of a product is equal to the product of the square roots. For example, √(a * b) = √a * √b.
• Real vs. Complex Numbers: The square root of a positive number is real. The square root of a negative number is imaginary.
• Positive and Negative Roots: For any positive number X, both √X and -√X are square roots of X.

These properties help in understanding the behavior and characteristics of the square root of 117 in various mathematical contexts.

Engage with these interactive examples to practice simplifying the square root of 117 and similar problems. These exercises will help reinforce the concepts and methods discussed.

Example 1: Simplifying the Square Root of 117

Let's start with the square root of 117. Follow these steps to simplify:

1. Find the prime factorization of 117: \(117 = 3^2 \times 13\).
2. Rewrite the square root: \(\sqrt{117} = \sqrt{3^2 \times 13}\).
3. Separate the factors: \(\sqrt{3^2} \times \sqrt{13} = 3\sqrt{13}\).

So, \(\sqrt{117}\) simplified is \(3\sqrt{13}\).

Example 2: Calculating Decimal Approximation

To find the decimal approximation of \(3\sqrt{13}\), use a calculator:

1. Calculate the square root of 13: \(\sqrt{13} \approx 3.605551275\).
2. Multiply by 3: \(3 \times 3.605551275 \approx 10.81665383\).

Thus, \(\sqrt{117} \approx 10.81665383\).

Interactive Problems

Now, try simplifying and approximating the following square roots:

• \(\sqrt{72}\)
• \(\sqrt{50}\)
• \(\sqrt{98}\)

Practice Quiz

Test your understanding with these questions:

1. What is the simplified form of \(\sqrt{72}\)?
2. Express \(\sqrt{50}\) in its simplest form.
3. Calculate the decimal approximation of \(\sqrt{98}\).

Check your answers:

1. \(\sqrt{72} = 6\sqrt{2}\)
2. \(\sqrt{50} = 5\sqrt{2}\)
3. \(\sqrt{98} \approx 9.899494937\)

Here are some frequently asked questions about the square root of 117:

• Can the square root of 117 be simplified?

  Yes, the square root of 117 can be simplified. It simplifies to \( 3\sqrt{13} \). This is because 117 can be factored into \( 9 \times 13 \), and the square root of 9 is 3. Therefore, \( \sqrt{117} = 3\sqrt{13} \).

• What is the simplified radical form of the square root of 117?

  The simplified radical form of the square root of 117 is \( 3\sqrt{13} \).

• Is the square root of 117 a rational or irrational number?

  The square root of 117 is an irrational number. This is because it cannot be expressed as a fraction of two integers. The decimal form of \( \sqrt{117} \) is non-terminating and non-repeating, which characterizes irrational numbers.

• Is 117 a perfect square?

  No, 117 is not a perfect square. A perfect square is a number that can be expressed as the square of an integer. Since 117 cannot be expressed as \( n^2 \) for any integer \( n \), it is not a perfect square.

• How can I approximate the square root of 117?

  The square root of 117 can be approximated using a calculator or long division method. In decimal form, \( \sqrt{117} \approx 10.8167 \). For practical purposes, it is often rounded to various decimal places, such as 10.82 for two decimal places.

• What are some properties of the square root of 117?

  The square root of 117, in its simplest form, is \( 3\sqrt{13} \). It is an irrational number and not a perfect square. It is used in various mathematical applications, especially in problems involving radicals and quadratic equations.

The square root of 117, approximately 10.816, can be utilized in various practical applications across different fields. Here are some examples:

• Geometry: The square root is often used to determine the length of the sides of geometric shapes. For example, if the area of a square room is 117 square feet, the length of each side would be the square root of 117, which is approximately 10.8 feet.
• Engineering: In civil engineering, the square root is used in calculations related to the natural frequency of structures such as bridges and buildings. Understanding these frequencies helps predict how structures will respond to loads and vibrations.
• Physics: In physics, square roots are used to compute various quantities such as the velocity of an object under certain conditions or the intensity of sound waves. For instance, the root mean square velocity in kinetic theory involves the square root of the average of squared velocities.
• Finance: In finance, the square root is used to calculate volatility in stock prices. The standard deviation, which is the square root of the variance, helps investors understand the risk associated with a particular stock.
• Statistics: Square roots are used to find standard deviations, which measure the dispersion of a dataset relative to its mean. This is crucial in fields such as economics and social sciences where data analysis is essential.
• Architecture: Architects use square roots to calculate dimensions and optimize materials. For example, the square root is involved in the Pythagorean theorem to determine the hypotenuse of right-angled triangles in structural designs.
• Navigation: In navigation, square roots are used to calculate distances between points on a map. For example, the distance between two points can be found using the Euclidean distance formula, which involves the square root of the sum of squared differences in coordinates.
• Computer Science: Square roots are employed in algorithms for encryption, image processing, and computer graphics. For instance, the distance formula in 2D and 3D graphics applications requires calculating the square root of the sum of squares of the differences in coordinates.

These examples demonstrate how the square root of 117, and square roots in general, are fundamental to many practical and theoretical applications in various disciplines.

Exploring related square roots can provide a deeper understanding of how different numbers are interconnected through their square roots. Here are some related square roots to 117:

• Square Root of 108: The square root of 108 can be simplified as \( \sqrt{108} = 6\sqrt{3} \). This is found by recognizing that 108 can be factored into \( 36 \times 3 \), and the square root of 36 is 6.
• Square Root of 121: The square root of 121 is a perfect square, \( \sqrt{121} = 11 \). This is because 121 can be expressed as \( 11 \times 11 \).
• Square Root of 144: Another perfect square, \( \sqrt{144} = 12 \). This is derived from the factorization \( 12 \times 12 \), making it a straightforward perfect square root.

Understanding these related square roots helps in recognizing patterns and simplifying other non-perfect squares like 117. By comparing these examples, we see that while 117 is not a perfect square and its simplified form is \( 3\sqrt{13} \), other numbers like 121 and 144 are perfect squares with whole number square roots.