How to Simplify the Square Root of 125: Easy Step-by-Step Guide

The process of simplifying the square root of 125 involves finding the prime factorization of the number and then simplifying the expression by grouping the prime factors. Let's go through the steps in detail:

Step-by-Step Simplification

1. Find the Prime Factorization of 125:

   The prime factors of 125 are found by dividing the number by the smallest prime numbers:

   • 125 is divisible by 5, so we divide:

   \[ 125 \div 5 = 25 \]

   • 25 is also divisible by 5, so we divide again:

   \[ 25 \div 5 = 5 \]

   • Finally, 5 is a prime number, so we stop here.

   Thus, the prime factorization of 125 is:

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

2. Group the Prime Factors:

   We can pair the prime factors in groups of two:

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

3. Take the Square Root:

   We apply the square root to the grouped factors:

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

   Using the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we get:

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

   Since \(\sqrt{5^2} = 5\), we can simplify the expression to:

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

Final Simplified Form

The simplified form of the square root of 125 is:

This means that the square root of 125 can be expressed as 5 times the square root of 5.

The concept of square roots is fundamental in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\).

Square roots are represented using the radical symbol \( \sqrt{} \). The number inside the radical is called the radicand. For instance, in \( \sqrt{125} \), 125 is the radicand.

Here's a brief overview of key points about square roots:

• Square roots can be both positive and negative, but in most contexts, we refer to the principal (positive) square root.
• Perfect squares are numbers whose square roots are integers (e.g., 1, 4, 9, 16).
• Non-perfect squares result in irrational numbers, which cannot be expressed as simple fractions.

To better understand square roots, let’s look at the process of simplifying \( \sqrt{125} \).

The square root of 125 is a value that, when multiplied by itself, gives the number 125. Mathematically, it is represented as √125. To simplify this, we need to find the simplest radical form.

Steps to Simplify √125:

1. Prime Factorization: Begin by finding the prime factors of 125. The prime factorization of 125 is:

   125 = 5 × 5 × 5 or 125 = 5³

2. Group the Factors: Since we are dealing with square roots, we look for pairs of prime factors. In this case, we have a pair of 5s (5²) and another 5:

   125 = (5²) × 5

3. Extract the Square Root: Apply the square root to each group of factors:

   √125 = √(5² × 5) = √(5²) × √5

4. Simplify: The square root of 5² is 5, so we can simplify further:

   √125 = 5 × √5

Thus, the simplified form of √125 is 5√5.

Verification:

To verify, you can square the simplified form and check if it equals 125:

(5√5)² = 5² × (√5)² = 25 × 5 = 125

Therefore, the simplified form 5√5 is correct.

By understanding these steps, you can simplify any similar square root problems efficiently.

Simplifying the square root of 125 involves breaking it down into its simplest radical form. Follow these detailed steps to understand the process:

1. Identify the Factors: Start by listing all the factors of 125.

   • Factors of 125: 1, 5, 25, 125

2. Find the Perfect Squares: From the list of factors, identify the perfect squares. A perfect square is a number that can be expressed as the product of an integer with itself.

   • Perfect squares in the list: 1, 25

3. Divide by the Largest Perfect Square: Choose the largest perfect square factor from the list. Here, it is 25. Divide 125 by 25.

   \[ \frac{125}{25} = 5 \]

4. Rewrite the Square Root: Express the original square root in terms of the identified perfect square.

   \[ \sqrt{125} = \sqrt{25 \times 5} \]

5. Separate the Square Root: Use the property of square roots that allows us to separate the product of two square roots.

   \[ \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} \]

6. Simplify the Square Roots: Simplify the square root of the perfect square.

   \[ \sqrt{25} = 5 \]

   Therefore,

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

7. Final Simplified Form: The simplified form of the square root of 125 is:

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

This step-by-step method can be applied to simplify other non-perfect square roots by identifying the largest perfect square factor and using the properties of square roots.

The prime factorization method is a straightforward approach to simplifying the square root of a number by breaking it down into its prime factors. Let's apply this method to the square root of 125.

1. Step 1: Find the Prime Factors of 125

   First, we need to express 125 as a product of prime numbers. We start by dividing 125 by the smallest prime number, which is 5.

   
   \(125 \div 5 = 25\)
   
   \(25 \div 5 = 5\)
   
   \(5 \div 5 = 1\)

   So, the prime factorization of 125 is:

   \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\)

2. Step 2: Group the Factors into Pairs

   Next, we group the prime factors in pairs. Since we are dealing with square roots, we look for pairs of identical factors:

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

3. Step 3: Simplify by Taking One Factor from Each Pair

   For each pair of prime factors, take one factor out of the square root sign:

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

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

\(\sqrt{125} = 5 \sqrt{5}\)

This method shows how breaking down a number into its prime factors can help simplify its square root effectively.

To simplify the square root of 125, it's essential to understand its factors. Here, we'll break down 125 into its prime factors and identify the perfect squares involved.

1. List the factors of 125:

   First, we need to list out all the factors of 125. These factors are numbers that multiply together to give 125:

   • 1, 5, 25, 125
2. Identify the perfect squares:

   Among the factors of 125, we need to find the perfect squares. A perfect square is a number that can be expressed as the square of an integer:

   • The perfect squares in our list of factors are: 1 and 25
3. Divide by the largest perfect square:

   To simplify the square root, we divide 125 by the largest perfect square identified in the previous step:

   • \( \frac{125}{25} = 5 \)
4. Calculate the square root of the perfect square:

   Next, we take the square root of the largest perfect square we found:

   • \( \sqrt{25} = 5 \)
5. Combine the results:

   Finally, we express the square root of 125 in its simplest form by combining the results from the division and the square root calculation:

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

Therefore, the simplified form of \( \sqrt{125} \) is \( 5\sqrt{5} \).

To simplify the square root of 125, we need to identify and extract perfect squares from its factors. This process makes the expression more manageable and provides a simpler form of the square root.

1. First, list the factors of 125:

   • 1
   • 5
   • 25
   • 125

2. Identify the perfect squares among these factors. A perfect square is a number that can be expressed as the square of an integer. From our list of factors, the perfect squares are:

   • 1
   • 25

3. Choose the largest perfect square factor, which is 25 in this case. This will help us in simplifying the radical more effectively.

4. Divide the original number (125) by the largest perfect square factor (25):

   \[
   125 \div 25 = 5
   \]

5. Now, express the original square root as a product of the square roots of the factors:

   \[
   \sqrt{125} = \sqrt{25 \times 5}
   \]

6. Using the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can simplify this further:

   \[
   \sqrt{125} = \sqrt{25} \times \sqrt{5}
   \]

7. Calculate the square root of the perfect square (25):

   \[
   \sqrt{25} = 5
   \]

8. Combine the results to obtain the simplified form:

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

By following these steps, we have successfully simplified the square root of 125 to \(5\sqrt{5}\).

Once you have simplified the square root of 125 to \( 5\sqrt{5} \), it's essential to verify that this result is correct. Follow these steps to ensure your simplification is accurate:

1. Rewrite the Original Expression:

   Start with the original square root expression: \( \sqrt{125} \).

2. Use the Simplified Form:

   We simplified \( \sqrt{125} \) to \( 5\sqrt{5} \). We will verify this by squaring \( 5\sqrt{5} \) and comparing it to the original value.

3. Square the Simplified Form:

   Calculate \( (5\sqrt{5})^2 \):

   • Square the coefficient: \( 5^2 = 25 \)
   • Square the radical: \( (\sqrt{5})^2 = 5 \)
   • Multiply these results: \( 25 \times 5 = 125 \)

   This shows that \( (5\sqrt{5})^2 = 125 \).

4. Compare with the Original Expression:

   Since \( (5\sqrt{5})^2 = 125 \) matches the original expression, our simplification is verified to be correct.

By following these steps, you can confidently confirm that \( 5\sqrt{5} \) is indeed the correct simplified form of \( \sqrt{125} \). Verifying your result is a useful habit to ensure accuracy in your mathematical work.

One alternative method for simplifying the square root of 125 is to use the prime factorization method. Another approach is to use the decimal approximation method.

• Prime Factorization Method: Find the prime factors of 125, which are 5 x 5 x 5. Since there is a set of three 5s, we can take one 5 out, leaving us with 5√5.
• Decimal Approximation Method: Use a calculator to find the square root of 125, which is approximately 11.18034.

When simplifying the square root of 125, avoid the following mistakes:

1. Confusing the square root of 125 with the square root of 25. Remember, the square root of 125 is not a whole number.
2. Forgetting to simplify the radical expression. Always look for perfect square factors that can be extracted.
3. Using incorrect prime factors. Make sure to factorize 125 correctly as 5 x 5 x 5.
4. Incorrectly applying the rules of exponents. Ensure you understand the properties of radicals and exponents.

Here are some practice problems involving the square root of 125 along with their solutions:

1. Simplify \( \sqrt{125} \).

Step 1: Factorize 125 as 5 x 5 x 5.	125 = 5 x 5 x 5
Step 2: Take one factor of 5 out of the radical.	√125 = 5√5
Step 3: Final simplified form is 5√5.

So, \( \sqrt{125} = 5\sqrt{5} \).

2. Simplify \( \sqrt{125} \) to the nearest whole number.

Step 1: Use a calculator to find the square root of 125.	√125 ≈ 11.18034
Step 2: Round the result to the nearest whole number.	√125 ≈ 11

Therefore, \( \sqrt{125} \) to the nearest whole number is 11.

The simplified square root of 125, which is \(5\sqrt{5}\), has various applications in mathematics and everyday life:

• Mathematics: The simplified form can be used in algebraic expressions and equations involving square roots, making calculations easier and more manageable.
• Geometry: In geometry, the simplified form can be used to calculate the side length of a square with area 125 square units, which is useful in various geometric problems.
• Engineering: Engineers often encounter square roots in calculations involving areas, volumes, and dimensions. The simplified form can help in making these calculations more efficient.
• Physics: In physics, the simplified form can be used in calculations related to waves, sound, and vibrations, where square roots commonly appear in formulas.
• Computer Science: In computer science, the simplified form can be used in algorithms and programming involving mathematical operations, simplifying code and improving efficiency.

In conclusion, the square root of 125 can be simplified to \(5\sqrt{5}\). This simplified form is useful in various mathematical and real-world applications, including algebra, geometry, engineering, physics, and computer science.

By understanding the methods for simplifying square roots, such as prime factorization and decimal approximation, you can efficiently simplify square roots and apply them in different contexts.