Simplified Square Root of 125: Easy Steps to Master the Concept

The simplified square root of 125 can be expressed in a more manageable form by factoring out the perfect square from the radicand.

Step-by-Step Simplification Process:

1. Identify the factors of 125 that include a perfect square:
   • 125 = 5 × 25
   • 25 is a perfect square (5²).
2. Rewrite the square root of 125 using the factors:
   • \(\sqrt{125} = \sqrt{5 \times 25}\)
3. Separate the square root into two parts:
   • \(\sqrt{125} = \sqrt{5} \times \sqrt{25}\)
4. Simplify the square root of the perfect square:
   • \(\sqrt{25} = 5\)
5. Combine the simplified parts:
   • \(\sqrt{125} = 5\sqrt{5}\)

Final Result:

The simplified form of the square root of 125 is:

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. Here, we will explore the basics of square roots, including their definition, properties, and examples.

Key concepts of square roots include:

• A square root of a number \( x \) is a number \( y \) such that \( y^2 = x \).
• The principal square root is the non-negative root and is denoted by the symbol \(\sqrt{}\).
• Every positive number has two square roots: one positive (principal) and one negative.

For example:

• \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).
• \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
• \(\sqrt{49} = 7\) because \(7 \times 7 = 49\).

Square roots can be simplified by factoring out perfect squares from the radicand. This process helps in reducing the square root to its simplest form, making it easier to work with.

For instance:

1. Identify the factors of the number that include a perfect square:
   • For 125, the factors are 5 and 25.
   • 25 is a perfect square (\(5^2\)).
2. Rewrite the square root using these factors:
   • \(\sqrt{125} = \sqrt{5 \times 25}\)
3. Simplify by separating the square root:
   • \(\sqrt{125} = \sqrt{5} \times \sqrt{25}\)
4. Simplify further by calculating the square root of the perfect square:
   • \(\sqrt{25} = 5\)
5. Combine the simplified parts:
   • \(\sqrt{125} = 5\sqrt{5}\)

Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. Recognizing and understanding perfect squares is essential in simplifying square roots and solving various mathematical problems.

Here are some key points about perfect squares:

• A perfect square is a number that can be written as \(n^2\), where \(n\) is an integer.
• For example, \(1, 4, 9, 16, 25,\) and \(36\) are all perfect squares.

Let's consider some examples of perfect squares:

• \(4 = 2^2\)
• \(9 = 3^2\)
• \(16 = 4^2\)
• \(25 = 5^2\)
• \(36 = 6^2\)

To identify a perfect square within a number, follow these steps:

1. Factorize the number into its prime factors.
   • For instance, the prime factors of 125 are \(5 \times 5 \times 5\).
2. Group the prime factors into pairs.
   • Here, \(125 = 5 \times 5 \times 5\) includes a pair of 5s, indicating a perfect square (5²).
3. Identify the largest perfect square that is a factor of the number.
   • For 125, the largest perfect square factor is \(25\) (since \(5^2 = 25\)).

Understanding perfect squares helps simplify square roots by allowing us to extract the square root of the perfect square factor. This makes the square root expression easier to handle and calculate.

For example, simplifying \(\sqrt{125}\) involves recognizing that 125 can be factored into \(5 \times 25\), and since 25 is a perfect square (\(5^2\)), we can simplify it as:

• \(\sqrt{125} = \sqrt{5 \times 25} = \sqrt{5} \times \sqrt{25} = \sqrt{5} \times 5 = 5\sqrt{5}\)

The prime factorization method is an effective way to simplify square roots by breaking down a number into its prime factors. This method helps identify and extract perfect square factors from the radicand, making it easier to simplify the square root.

Here are the steps to use the prime factorization method to simplify the square root of 125:

1. Find the prime factors of 125:
   • Start by dividing 125 by the smallest prime number, which is 5.
   • \(125 \div 5 = 25\)
   • Next, divide 25 by 5 (since 5 is still a prime number).
   • \(25 \div 5 = 5\)
   • Finally, divide 5 by 5.
   • \(5 \div 5 = 1\)
2. Write 125 as a product of its prime factors:
   • \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\)
3. Identify pairs of prime factors to find perfect squares:
   • In \(125 = 5^3\), we have one pair of 5s (since \(5^2 = 25\)).
4. Rewrite the square root of 125 using the prime factors:
   • \(\sqrt{125} = \sqrt{5^3} = \sqrt{5^2 \times 5}\)
5. Separate the perfect square from the non-perfect square factor:
   • \(\sqrt{125} = \sqrt{5^2} \times \sqrt{5}\)
6. Simplify the square root of the perfect square:
   • \(\sqrt{5^2} = 5\)
7. Combine the simplified parts:
   • \(\sqrt{125} = 5\sqrt{5}\)

By using the prime factorization method, we have successfully simplified the square root of 125 to \(5\sqrt{5}\). This method can be applied to any number to simplify its square root by identifying and extracting perfect square factors.

The square root of 125 can be simplified using the prime factorization method. Here's a detailed step-by-step process:

1. Prime Factorization:

   First, we find the prime factors of 125.

   125 can be written as a product of prime factors:

   \[ 125 = 5 \times 5 \times 5 = 5^3 \]

2. Group the Factors:

   Next, we group the prime factors in pairs. Since we are finding the square root, we look for pairs of identical factors:

   \[ 125 = (5 \times 5) \times 5 = 5^2 \times 5 \]

3. Apply the Square Root:

   We can now take the square root of each group of factors:

   \[ \sqrt{125} = \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5} \]

4. Simplify:

   Since the square root of \(5^2\) is 5, we can simplify further:

   \[ \sqrt{125} = 5 \times \sqrt{5} = 5\sqrt{5} \]

Thus, the simplified form of the square root of 125 is \( 5\sqrt{5} \).

In decimal form, \( 5\sqrt{5} \approx 11.1803398875 \).

This method shows that while 125 is not a perfect square, its square root can be expressed in a simplified radical form.

Using these steps, you can simplify other square roots in a similar manner by finding their prime factors, grouping them, and then applying the square root.

The process of simplifying the square root of 125 involves breaking down the number into its prime factors and using those factors to find the simplest radical form.

1. Prime Factorization:

   First, we find the prime factors of 125. This can be expressed as:

   \[
   125 = 5 \times 5 \times 5 = 5^3
   \]

2. Grouping Factors:

   Next, we group the factors in pairs of two. Each pair of 5s will come out of the square root as a single 5:

   \[
   \sqrt{125} = \sqrt{5^2 \times 5} = 5\sqrt{5}
   \]

3. Simplifying the Expression:

   Since we have extracted one pair of 5s out of the square root, the simplest form of the square root of 125 is:

   \[
   \sqrt{125} = 5\sqrt{5}
   \]

4. Decimal Approximation:

   For a decimal approximation, we can calculate:

   \[
   5\sqrt{5} \approx 5 \times 2.236 = 11.180
   \]

Thus, the simplified form of the square root of 125 is \(5\sqrt{5}\), and its decimal approximation is approximately 11.180.

The square root of 125 can be simplified using the method of prime factorization. Here's a detailed mathematical explanation:

1. Prime Factorization:

   First, find the prime factors of 125. We get:

   \[ 125 = 5^3 = 5 \times 5 \times 5 \]

2. Group the Factors:

   We group the factors in pairs of two identical numbers:

   \[ 125 = 5^2 \times 5 \]

3. Apply the Square Root:

   Next, take the square root of the groups:

   \[ \sqrt{125} = \sqrt{5^2 \times 5} \]

   According to the property of square roots:

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

4. Simplify the Expression:

   We simplify the expression by taking the square root of 5²:

   \[ \sqrt{125} = \sqrt{5^2} \times \sqrt{5} = 5 \times \sqrt{5} \]

   Thus, the simplified form of the square root of 125 is:

   \[ \sqrt{125} = 5\sqrt{5} \]

5. Decimal Form:

   For a decimal approximation, we can further evaluate:

   \[ 5\sqrt{5} \approx 5 \times 2.236 = 11.18 \]

This process breaks down the steps involved in simplifying the square root of 125 to \( 5\sqrt{5} \), making it easier to understand and apply in mathematical problems.

The simplified square root of 125 can be represented visually to enhance understanding. Here's a step-by-step visual breakdown:

Prime Factorization

First, we find the prime factors of 125:

• 125 = 5 × 5 × 5

This can be shown as:

125 = 5² × 5

Simplifying the Square Root

Next, we apply the square root to the prime factorization:

• \(\sqrt{125} = \sqrt{5^2 \times 5}\)

Breaking this down visually:

• \(\sqrt{5^2}\) can be simplified to 5
• The remaining factor is \(\sqrt{5}\)

Combining these, we get:

• \(\sqrt{125} = 5\sqrt{5}\)

Graphical Illustration

Consider a square with an area of 125 square units. To find the side length (the square root), we simplify it to:

• \(\sqrt{125} = 5\sqrt{5}\)

This shows that the side length of the square can be visualized as 5 units times the square root of 5 units.

Geometric Representation

Imagine breaking down the area into smaller, more manageable squares:

• One large square of 25 units (5 × 5)
• Five smaller units, each representing \(\sqrt{5}\)

Thus, visually representing the square root of 125 involves combining these units to form the side length:

This diagram helps to conceptualize the simplification process.

Simplified square roots play a crucial role in various fields due to their ability to make complex calculations more manageable. Here are some practical applications of simplified square roots:

• Finance:

  In finance, square roots are used to calculate the volatility of stock prices. The standard deviation, which measures the variability of stock returns, involves taking the square root of the variance. This helps investors assess the risk associated with different investments.

• Engineering:

  Engineers use square roots to determine the natural frequency of structures like bridges and buildings. This helps predict how structures will respond to various loads and ensure their stability and safety.

• Science:

  Square roots are used in scientific calculations such as determining the velocity of moving objects, the absorption of radiation, and the intensity of sound waves. These calculations are essential for understanding natural phenomena and developing new technologies.

• Statistics:

  In statistics, the square root is used to calculate the standard deviation, which indicates how much a set of data deviates from the mean. This is important for data analysis and making informed decisions based on statistical results.

• Geometry:

  Square roots are fundamental in geometry, especially in the Pythagorean theorem, which involves calculating the lengths of the sides of right triangles. This theorem is used to solve various geometric problems.

• Computer Science:

  In computer programming, square roots are used in algorithms for encryption, image processing, and game physics. For example, encryption algorithms often involve modular arithmetic and square roots to secure data transmissions.

• Navigation:

  Pilots and navigators use square roots to calculate distances between points on a map and to estimate directions. This is crucial for accurate navigation and planning routes.

• Electrical Engineering:

  Square roots are used to compute power, voltage, and current in electrical circuits. These calculations are essential for designing and developing electrical systems and devices.

• Photography:

  The aperture of a camera lens, which controls the amount of light entering the camera, is related to the f-number. The area of the aperture is proportional to the square of the f-number, affecting the exposure of photographs.

These examples illustrate the diverse and critical applications of simplified square roots in various fields, highlighting their importance in both theoretical and practical contexts.

Understanding the simplification of the square root of 125 can be further reinforced by looking at similar examples. Below are several examples of square roots that have been simplified using the same techniques:

Example 1: Simplifying √50

1. Identify the prime factors of 50: \( 50 = 2 \times 5 \times 5 \).
2. Group the prime factors into pairs: \( 50 = (5 \times 5) \times 2 \).
3. Apply the square root to each pair: \( \sqrt{50} = \sqrt{5^2 \times 2} = 5\sqrt{2} \).

Example 2: Simplifying √72

1. Identify the prime factors of 72: \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \).
2. Group the prime factors into pairs: \( 72 = (2 \times 2) \times (3 \times 3) \times 2 \).
3. Apply the square root to each pair: \( \sqrt{72} = \sqrt{2^2 \times 3^2 \times 2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} \).

Example 3: Simplifying √45

1. Identify the prime factors of 45: \( 45 = 3 \times 3 \times 5 \).
2. Group the prime factors into pairs: \( 45 = (3 \times 3) \times 5 \).
3. Apply the square root to each pair: \( \sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5} \).

Example 4: Simplifying √18

1. Identify the prime factors of 18: \( 18 = 2 \times 3 \times 3 \).
2. Group the prime factors into pairs: \( 18 = 2 \times (3 \times 3) \).
3. Apply the square root to each pair: \( \sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2} \).

Example 5: Simplifying √98

1. Identify the prime factors of 98: \( 98 = 2 \times 7 \times 7 \).
2. Group the prime factors into pairs: \( 98 = 2 \times (7 \times 7) \).
3. Apply the square root to each pair: \( \sqrt{98} = \sqrt{2 \times 7^2} = 7\sqrt{2} \).

By practicing these examples, you can gain a deeper understanding of the simplification process and apply it to other numbers effectively.

To help reinforce your understanding of simplifying square roots, try solving the following practice problems. Each problem involves simplifying the square root of a number, similar to the square root of 125. Make sure to show all your steps clearly.

1. Simplify the square root of 50:

   \(\sqrt{50}\)

   Solution:

   • Step 1: Factor 50 into its prime factors: \(50 = 2 \times 5^2\).
   • Step 2: Rewrite the square root: \(\sqrt{50} = \sqrt{2 \times 5^2}\).
   • Step 3: Take the square root of the perfect square: \(5\sqrt{2}\).
   • Answer: \(5\sqrt{2}\).

2. Simplify the square root of 72:

   \(\sqrt{72}\)

   Solution:

   • Step 1: Factor 72 into its prime factors: \(72 = 2^3 \times 3^2\).
   • Step 2: Rewrite the square root: \(\sqrt{72} = \sqrt{2^3 \times 3^2}\).
   • Step 3: Take the square root of the perfect squares: \(3 \sqrt{8}\).
   • Answer: \(6\sqrt{2}\).

3. Simplify the square root of 98:

   \(\sqrt{98}\)

   Solution:

   • Step 1: Factor 98 into its prime factors: \(98 = 2 \times 7^2\).
   • Step 2: Rewrite the square root: \(\sqrt{98} = \sqrt{2 \times 7^2}\).
   • Step 3: Take the square root of the perfect square: \(7 \sqrt{2}\).
   • Answer: \(7\sqrt{2}\).

4. Simplify the square root of 200:

   \(\sqrt{200}\)

   Solution:

   • Step 1: Factor 200 into its prime factors: \(200 = 2^3 \times 5^2\).
   • Step 2: Rewrite the square root: \(\sqrt{200} = \sqrt{2^3 \times 5^2}\).
   • Step 3: Take the square root of the perfect squares: \(10 \sqrt{2}\).
   • Answer: \(10\sqrt{2}\).

5. Simplify the square root of 18:

   \(\sqrt{18}\)

   Solution:

   • Step 1: Factor 18 into its prime factors: \(18 = 2 \times 3^2\).
   • Step 2: Rewrite the square root: \(\sqrt{18} = \sqrt{2 \times 3^2}\).
   • Step 3: Take the square root of the perfect square: \(3 \sqrt{2}\).
   • Answer: \(3\sqrt{2}\).

Practice solving these problems to become more comfortable with the process of simplifying square roots. Once you have mastered these, try creating your own problems or find additional exercises to continue practicing.

The simplified square root of 125 is represented as \( \sqrt{125} = 5\sqrt{5} \).

This simplification process involves identifying the perfect square factors of 125, which are 25 and 5. Breaking down 125 into its prime factors (5 × 5 × 5) helps in expressing the square root in its simplified form. By extracting the square root of 25 and leaving 5 inside the square root symbol, we arrive at \( 5\sqrt{5} \).

Understanding and applying this method not only simplifies the expression but also enhances mathematical problem-solving skills related to square roots and prime factorization.

What is the simplified square root of 125?

The simplified square root of 125 is \( 5\sqrt{5} \).

How do you simplify \( \sqrt{125} \)?

To simplify \( \sqrt{125} \), you identify the perfect square factors within 125, which are 25 and 5. This leads to \( \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} \).

Why is it important to simplify square roots?

Simplifying square roots helps in making mathematical expressions more manageable and understandable. It also facilitates further calculations and applications in various fields of mathematics and sciences.

Can the square root of 125 be simplified further?

No, \( \sqrt{125} \) is already in its simplest form as \( 5\sqrt{5} \).

What are some applications of understanding simplified square roots?

Understanding simplified square roots is essential in fields like engineering, physics, and mathematics where calculations involving square roots are frequent. It simplifies equations, aids in problem-solving, and enhances comprehension of mathematical concepts.