Simplify Square Root 125: Easy Steps to Simplify √125

The process of simplifying the square root of 125 involves breaking down the number into its prime factors and then simplifying the radical expression.

Step-by-Step Process

1. First, factorize 125 into its prime factors.
• 125 = 5 × 25
• Therefore, 125 = 5 × 5 × 5 or 5³
2. Next, express the square root of 125 using these factors.
• \(\sqrt{125} = \sqrt{5^3}\)
3. Since the square root of a product is the product of the square roots, and knowing that \(\sqrt{a^2} = a\), we can simplify:
• \(\sqrt{5^3} = \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5} = 5\sqrt{5}\)

Thus, the simplified form of \(\sqrt{125}\) is:

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

Conclusion

By breaking down the number 125 into its prime factors and using the properties of square roots, we find that the square root of 125 simplifies to \(5\sqrt{5}\).

The process of simplifying the square root of 125 involves breaking down the number into its prime factors and simplifying the radical expression. Understanding the steps to achieve this can help in solving various mathematical problems efficiently. In this section, we will explore the step-by-step method to simplify √125.

• Prime Factorization: Identify the prime factors of 125.
• Grouping Factors: Group the factors in pairs of the same numbers.
• Extracting Factors: Extract one factor from each group to simplify the radical.
• Final Simplification: Combine the extracted factors to get the simplified form.

By following these steps, the square root of 125 can be simplified to 5√5, making it easier to work with in further calculations.

Square roots are a fundamental concept in mathematics, often introduced in early algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 * 5 = 25. Square roots are denoted by the radical symbol √.

The process of simplifying square roots involves expressing the number inside the square root as a product of its prime factors and then simplifying it. For example, to simplify the square root of 125, we follow these steps:

1. Find the prime factors of 125. Since 125 = 5 × 5 × 5, we can write it as 125 = 5² × 5.
2. Apply the property of square roots that √(a × b) = √a × √b.
3. Simplify the expression: √(5² × 5) = √(5²) × √5 = 5√5.

Thus, the simplified form of the square root of 125 is 5√5. Understanding this process helps in solving more complex mathematical problems and is crucial for students learning algebra and higher-level math.

To simplify the square root of 125 using the prime factorization method, follow these steps:

1. Find the prime factors of 125.

125 can be factored into prime numbers: \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\).

2. Group the prime factors in pairs.

In \(5^3\), we can group two of the 5s: \(125 = (5^2) \times 5\).

3. Take the square root of each group of prime factors.

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

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

To simplify the square root of 125, we start by breaking down the number into its prime factors. Here’s a step-by-step guide:

1. List the factors: The factors of 125 are 1, 5, 25, and 125.

2. Identify the prime factors: 125 can be expressed as the product of its prime factors:

   \[ 125 = 5 \times 25 \]

   Since 25 is also a perfect square, we further break it down:

   \[ 25 = 5 \times 5 \]

   So, \( 125 = 5 \times 5 \times 5 \) or \( 5^3 \).

3. Rewrite under the radical: Now we rewrite the square root of 125 using these factors:

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

4. Separate the terms: Use the property of square roots to separate the terms:

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

5. Simplify: Take the square root of the perfect square \( 5^2 \):

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

   Thus, we have:

   \[ 5 \sqrt{5} \]

Therefore, the simplest radical form of \( \sqrt{125} \) is:

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

The properties of square roots are fundamental in simplifying and understanding various mathematical operations. Here are some key properties:

• Product Rule: The square root of a product is equal to the product of the square roots of the factors.
  Example: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)
• Quotient Rule: The square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator.
  Example: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
• Square of a Square Root: The square of a square root returns the original value.
  Example: \( (\sqrt{a})^2 = a \)
• Square Root of a Square: The square root of a squared number gives the absolute value of the original number.
  Example: \( \sqrt{a^2} = |a| \)
• Sum and Difference: The square root of a sum or difference of squares does not simplify to the sum or difference of the square roots.
  Example: \( \sqrt{a^2 + b^2} \neq \sqrt{a^2} + \sqrt{b^2} \)
• Irrational Numbers: The square root of a non-perfect square is an irrational number, which cannot be expressed as a simple fraction.
  Example: \( \sqrt{2} \) is an irrational number.

These properties are useful in simplifying complex expressions and solving equations involving square roots.

Simplifying radicals involves breaking down the number under the radical sign into its prime factors and simplifying wherever possible. Here, we'll take a detailed look at simplifying the square root of 125 as an example.

Step-by-Step Process

1. Find the prime factors of the number under the radical.

Prime factorization of 125:

125 = 5 × 25 = 5 × (5 × 5) = 5³

2. Group the prime factors in pairs.

Since we are dealing with a square root, we group the factors into pairs of two. For 125, this gives us:

125 = 5² × 5

3. Move each pair of prime factors out from under the radical.

We can take the square root of 5² which is 5, and move it outside the radical:

√(5² × 5) = 5√5

4. Simplify the expression.

There are no more pairs of prime factors under the radical, so the simplified form of √125 is:

√125 = 5√5

Key Points

• The square root of a product is the product of the square roots of each factor.
• Identify and extract pairs of prime factors from under the radical sign.
• Leave any factors that don't form pairs under the radical sign.

By following these steps, you can simplify any square root expression by breaking it down into its prime factors and extracting pairs to simplify the radical.

Simplifying the square root of 125 involves breaking down the number into its prime factors and then simplifying the radical. Here are the detailed steps:

1. List the factors of 125:

   • 1, 5, 25, 125

2. Identify the perfect squares from the list of factors:

   • 1, 25

3. Divide 125 by the largest perfect square:

   • \( \frac{125}{25} = 5 \)

4. Calculate the square root of the largest perfect square:

   • \( \sqrt{25} = 5 \)

5. Combine the results to get the simplified form:

   • \( \sqrt{125} = 5\sqrt{5} \)

Thus, the square root of 125 in its simplest radical form is \( 5\sqrt{5} \).

To simplify the square root of 125, we will break down the process into clear steps using the prime factorization method. This method helps in understanding how to simplify any square root.

1. Find the Prime Factorization: First, we need to find the prime factors of 125. The prime factorization of 125 is:

   125 = 5 × 5 × 5 = 5³

2. Group the Factors: Group the prime factors in pairs. For 125, we can group two of the 5s together:

   5³ = (5 × 5) × 5 = 5² × 5

3. Take the Square Root of Each Group: Take the square root of the pairs of factors and simplify. The square root of 5² is 5:

   √(5² × 5) = 5 × √5

4. Simplify the Expression: Finally, write the simplified form. Therefore, the square root of 125 simplifies to:

   √125 = 5√5

This step-by-step approach can be applied to simplify any square root. It involves breaking down the number into its prime factors, grouping them into pairs, and then simplifying.

Let's summarize the steps:

• Find the prime factorization of the number.
• Group the factors into pairs.
• Take the square root of each group of pairs.
• Simplify the expression.

When simplifying the square root of 125, there are several common mistakes that students often make. Understanding these mistakes can help ensure accurate simplification and prevent errors. Here are some of the most frequent pitfalls:

• Ignoring Perfect Squares: One of the most common mistakes is failing to recognize and factor out perfect squares from the radicand. For example, in √125, the number 25 is a perfect square (25 = 5^2). The correct simplification involves recognizing this and breaking down √125 into √(25 * 5), which simplifies to 5√5.
• Rushing Through Steps: Simplifying radicals requires careful step-by-step work. Rushing can lead to skipped steps or incorrect factorization. Always take your time to factor the radicand completely before simplifying.
• Incorrect Factorization: Incorrectly factoring the radicand can lead to the wrong simplified form. Ensure that the factors used are correct and that any perfect squares are properly identified and extracted.
• Forgetting to Simplify Completely: After initial simplification, always double-check to see if further simplification is possible. For instance, if after simplifying √125 to 5√5, check if √5 can be simplified further (it cannot in this case, but always verify).
• Misinterpreting the Radical Symbol: Confusing the properties of square roots with other operations can lead to errors. Remember that √(a * b) = √a * √b, and use this property correctly during simplification.

By being mindful of these common mistakes, you can enhance your accuracy in simplifying square roots and avoid common pitfalls.

Practicing the simplification of square roots is essential for mastering this concept. Here are some practice problems to help you become more proficient:

1. Problem: Simplify \(\sqrt{72}\)
   1. Step 1: Factor 72 into its prime factors: \(72 = 2^3 \times 3^2\)
   2. Step 2: Pair the prime factors: \(2^3 = 2^2 \times 2\) and \(3^2\)
   3. Step 3: Take the square root of the paired factors: \(\sqrt{2^2} = 2\) and \(\sqrt{3^2} = 3\)
   4. Step 4: Combine the results: \(2 \times 3 \times \sqrt{2} = 6\sqrt{2}\)

   Answer: \(6\sqrt{2}\)

2. Problem: Simplify \(\sqrt{98}\)
   1. Step 1: Factor 98 into its prime factors: \(98 = 2 \times 7^2\)
   2. Step 2: Pair the prime factors: \(7^2\)
   3. Step 3: Take the square root of the paired factor: \(\sqrt{7^2} = 7\)
   4. Step 4: Combine the results: \(7\sqrt{2}\)

   Answer: \(7\sqrt{2}\)

3. Problem: Simplify \(\sqrt{200}\)
   1. Step 1: Factor 200 into its prime factors: \(200 = 2^3 \times 5^2\)
   2. Step 2: Pair the prime factors: \(2^2\) and \(5^2\)
   3. Step 3: Take the square root of the paired factors: \(\sqrt{2^2} = 2\) and \(\sqrt{5^2} = 5\)
   4. Step 4: Combine the results: \(2 \times 5 \times \sqrt{2} = 10\sqrt{2}\)

   Answer: \(10\sqrt{2}\)

4. Problem: Simplify \(\sqrt{50}\)
   1. Step 1: Factor 50 into its prime factors: \(50 = 2 \times 5^2\)
   2. Step 2: Pair the prime factors: \(5^2\)
   3. Step 3: Take the square root of the paired factor: \(\sqrt{5^2} = 5\)
   4. Step 4: Combine the results: \(5\sqrt{2}\)

   Answer: \(5\sqrt{2}\)

5. Problem: Simplify \(\sqrt{180}\)
   1. Step 1: Factor 180 into its prime factors: \(180 = 2^2 \times 3^2 \times 5\)
   2. Step 2: Pair the prime factors: \(2^2\) and \(3^2\)
   3. Step 3: Take the square root of the paired factors: \(\sqrt{2^2} = 2\) and \(\sqrt{3^2} = 3\)
   4. Step 4: Combine the results: \(2 \times 3 \times \sqrt{5} = 6\sqrt{5}\)

   Answer: \(6\sqrt{5}\)

These practice problems cover a range of difficulties and help reinforce the steps needed to simplify square roots effectively. Keep practicing to build confidence and proficiency!

• What is the square root of 125?

  The square root of 125 is equal to \(5\sqrt{5}\) or approximately 11.18.

• Is 125 a perfect square?

  No, 125 is not a perfect square. It is a perfect cube.

• Is \(\sqrt{125}\) a rational or irrational number?

  \(\sqrt{125}\) is an irrational number because its simplified form \(5\sqrt{5}\) results in a non-terminating and non-repeating decimal value.

• What is the simplest form of the square root of 125?

  The simplest form of the square root of 125 is \(5\sqrt{5}\).

• How can we find the square root of 125?

  We can find the square root of 125 using either the prime factorization method or the long division method.

• What are the properties of the square root of 125?

  The square root of 125 has several properties:
  

  
  
  • It is an irrational number.
  
  
  • It is not a whole number.
  
  
  • It can be expressed as \(5\sqrt{5}\).
  
  
  • It is approximately equal to 11.18.