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

The square root of 125 can be simplified by expressing 125 as a product of its prime factors.

Step-by-Step Simplification

1. First, factorize 125 into its prime factors:
   • 125 = 5 × 25
   • So, 125 = 5 × 5 × 5 or 125 = 5³
2. Next, apply the square root to the prime factorization:
   • \(\sqrt{125} = \sqrt{5^3}\)
3. Rewrite the expression to separate the perfect square from the non-perfect square:
   • \(\sqrt{5^3} = \sqrt{5^2 \cdot 5} = \sqrt{5^2} \cdot \sqrt{5}\)
4. Simplify the square root of the perfect square:
   • \(\sqrt{5^2} = 5\)
5. Combine the results:
   • \(\sqrt{125} = 5\sqrt{5}\)

Conclusion

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

Simplifying square roots is an essential skill in algebra that helps to make complex expressions more manageable. The process involves breaking down a square root into its simplest form. Let's begin with a step-by-step approach to simplifying the square root of 125.

1. Understanding Square Roots: The 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 \(3 \times 3 = 9\).
2. Prime Factorization: To simplify a square root, we first need to factorize the number into its prime factors. Prime factorization involves breaking down a number into the product of prime numbers.
   • For 125, we start by dividing by the smallest prime number, which is 5: \(125 \div 5 = 25\).
   • Next, we factor 25: \(25 \div 5 = 5\).
   • So, \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\).
3. Simplifying the Square Root: With the prime factorization, we can now simplify the square root.
   • Write the square root with the prime factors: \(\sqrt{125} = \sqrt{5^3}\).
   • Separate the perfect square from the non-perfect square: \(\sqrt{5^3} = \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5}\).
   • Simplify the square root of the perfect square: \(\sqrt{5^2} = 5\).
   • Combine the results: \(\sqrt{125} = 5\sqrt{5}\).

By following these steps, you can simplify the square root of 125 to \(5\sqrt{5}\). This method can be applied to other square roots, helping you to simplify and understand them better.

Square roots are a fundamental concept in mathematics that are crucial for simplifying expressions and solving equations. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \(3 \times 3 = 9\). In mathematical terms, the square root of a number \(x\) is written as \(\sqrt{x}\).

Here is a detailed step-by-step explanation to understand square roots better:

1. Definition:
   • A square root of a number \(x\) is a number \(y\) such that \(y^2 = x\).
   • For example, \(\sqrt{16} = 4\) because \(4^2 = 16\).
2. Principal Square Root:
   • The principal square root of a non-negative number \(x\) is the non-negative value \(y\) such that \(y^2 = x\).
   • It is denoted as \(\sqrt{x}\) and is always non-negative.
3. Properties of Square Roots:
   • \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
   • \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
   • \((\sqrt{a})^2 = a\)
   • \(\sqrt{a^2} = |a|\) (the absolute value of \(a\))
4. Perfect Squares:
   • A perfect square is an integer that is the square of another integer.
   • Examples include \(1, 4, 9, 16, 25, 36, \ldots\)
5. Non-Perfect Squares:
   • Numbers that are not perfect squares have irrational square roots.
   • For example, \(\sqrt{2}\) and \(\sqrt{3}\) are irrational numbers.

Understanding these basic concepts of square roots will help you simplify square roots of more complex numbers, such as 125, and make algebraic operations easier and more intuitive.

The prime factorization method is a systematic approach to simplify square roots by breaking down the number into its prime factors. This method helps in identifying perfect squares that can be easily simplified. Let's explore the prime factorization method to simplify the square root of 125.

1. Identify Prime Factors:
   • Prime factorization involves expressing a number as a product of prime numbers.
   • Start by dividing the number by the smallest prime number, which is 2, and continue dividing by prime numbers until you reach a prime number.
2. Factorize 125:
   • 125 is not divisible by 2 (the smallest prime number), so we move to the next prime number, which is 3. 125 is also not divisible by 3.
   • The next prime number is 5. 125 is divisible by 5: \(125 \div 5 = 25\).
   • Continue factorizing 25: \(25 \div 5 = 5\).
   • So, \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\).
3. Simplify the Square Root:
   • Write the square root with the prime factors: \(\sqrt{125} = \sqrt{5^3}\).
   • Separate the perfect square from the non-perfect square: \(\sqrt{5^3} = \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5}\).
   • Simplify the square root of the perfect square: \(\sqrt{5^2} = 5\).
   • Combine the results: \(\sqrt{125} = 5\sqrt{5}\).

By following these steps, the square root of 125 is simplified to \(5\sqrt{5}\). The prime factorization method can be applied to any number, making it a versatile tool for simplifying square roots.

Simplifying the square root of 125 involves a systematic approach using prime factorization and identifying perfect squares. Follow these detailed steps to simplify \(\sqrt{125}\).

1. Prime Factorization:
   • First, factorize 125 into its prime factors.
   • Divide 125 by the smallest prime number. Since 125 is not divisible by 2 or 3, try 5:
   • \(125 \div 5 = 25\).
   • Next, factorize 25: \(25 \div 5 = 5\).
   • So, \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\).
2. Rewrite the Square Root:
   • Express the square root of 125 using its prime factors:
   • \(\sqrt{125} = \sqrt{5^3}\).
3. Separate Perfect Squares:
   • Identify and separate the perfect squares from the remaining factors:
   • \(\sqrt{5^3} = \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5}\).
4. Simplify the Perfect Square:
   • Simplify the square root of the perfect square:
   • \(\sqrt{5^2} = 5\).
5. Combine the Results:
   • Multiply the simplified square root of the perfect square by the remaining square root:
   • \(\sqrt{125} = 5 \times \sqrt{5} = 5\sqrt{5}\).

By following these steps, the square root of 125 is simplified to \(5\sqrt{5}\). This method can be applied to other square roots, making the process of simplification straightforward and efficient.

While the prime factorization method is the most common approach to simplifying square roots, there are alternative methods that can also be effective. Let's explore some of these methods for simplifying the square root of 125.

1. Using the Approximation Method:
   • Find two perfect squares between which 125 lies. These are 121 (\(11^2\)) and 144 (\(12^2\)).
   • Estimate the square root by finding a value between 11 and 12. Since 125 is closer to 121, \(\sqrt{125}\) is slightly more than 11.
   • This method provides an approximate value: \(\sqrt{125} \approx 11.18\).
2. Using the Factorization Method:
   • Break down the number into its factors and simplify incrementally.
   • Factor 125 as \(25 \times 5\).
   • Simplify the square root of each factor separately: \(\sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5 \times \sqrt{5} = 5\sqrt{5}\).
3. Using Simplification Rules:
   • Apply general rules for simplifying square roots.
   • For any number \(a\) and \(b\), \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
   • Apply this rule to 125: \(\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}\).
4. Using Algebraic Manipulation:
   • Express the number under the square root as a product of squares and other factors.
   • Write 125 as \(5^3\) and then use properties of exponents and roots: \(\sqrt{5^3} = \sqrt{(5^2 \times 5)} = \sqrt{5^2} \times \sqrt{5} = 5\sqrt{5}\).

These alternative methods offer different approaches to simplifying the square root of 125, making it easier to understand and apply the concept in various contexts.

When simplifying square roots, it is easy to make mistakes that can lead to incorrect results. Here are some common mistakes to avoid when simplifying the square root of 125.

1. Incorrect Prime Factorization:
   • Ensure that the number is correctly factorized into its prime factors.
   • For example, misidentifying the prime factors of 125 can lead to errors. The correct factorization is \(125 = 5 \times 5 \times 5\) or \(125 = 5^3\).
2. Forgetting to Simplify Perfect Squares:
   • Always look for and simplify perfect square factors within the square root.
   • For instance, failing to recognize that \(25\) is a perfect square within \(\sqrt{125}\) and simplifying it as \(\sqrt{25} = 5\) can lead to incorrect results.
3. Incorrect Application of Square Root Properties:
   • Ensure the correct use of properties of square roots, such as \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
   • Mistaking this property can lead to incorrect simplifications, such as \(\sqrt{125} = \sqrt{5^3}\) should be \(\sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5} = 5\sqrt{5}\).
4. Omitting Intermediate Steps:
   • Skipping steps can cause confusion and errors in the final result.
   • Ensure to break down the process step-by-step: factorize, separate perfect squares, simplify, and combine results.
5. Rounding Too Early:
   • If approximating the square root, avoid rounding numbers too early in the process.
   • For example, approximating \(\sqrt{125}\) too early can lead to less accurate results. Simplify first to \(5\sqrt{5}\) before approximating if needed.

By avoiding these common mistakes, you can ensure the correct and accurate simplification of the square root of 125, resulting in \(5\sqrt{5}\).

Simplifying square roots is not just an academic exercise; it has practical applications in various fields. Understanding how to simplify square roots can be useful in real-world scenarios. Here are some practical applications:

1. Geometry and Trigonometry:
   • Square roots often appear in geometric formulas, such as calculating the diagonal of a square or the hypotenuse of a right triangle.
   • For example, the diagonal \(d\) of a square with side length \(a\) is given by \(d = \sqrt{2a^2} = a\sqrt{2}\).
   • Simplifying square roots makes it easier to work with these formulas and understand geometric relationships.
2. Physics:
   • Square roots are frequently used in physics equations, such as those involving acceleration, force, and energy.
   • For instance, in the formula for the period of a pendulum \(T = 2\pi \sqrt{\frac{L}{g}}\), where \(L\) is the length and \(g\) is the acceleration due to gravity, simplifying the square root can help in precise calculations.
3. Engineering:
   • Engineering problems often involve square roots, particularly in calculations of stress, strain, and electrical circuits.
   • Simplifying these square roots can make complex equations more manageable and easier to solve.
4. Finance:
   • Square roots are used in financial models, such as calculating the standard deviation of investment returns.
   • Simplifying square roots helps in understanding and analyzing the volatility and risk of investments.
5. Computer Science:
   • Algorithms often involve square roots, especially in areas like computer graphics, machine learning, and data analysis.
   • Efficiently simplifying square roots can optimize these algorithms and improve computational performance.
6. Everyday Life:
   • Simplifying square roots can also be useful in daily activities, such as measuring distances, calculating areas, and understanding proportions.
   • For example, when determining the side length of a square with a given area, simplifying the square root helps in making quick and accurate measurements.

By mastering the simplification of square roots, you can apply this knowledge to various practical situations, enhancing your problem-solving skills and mathematical understanding.

Here are some practice problems and solutions to help you master the simplification of square roots. Follow the step-by-step solutions to understand the process.

Problem 1: Simplify the Square Root of 125

1. Prime Factorization Method:
   • Step 1: Find the prime factors of 125.
   • \(125 = 5 \times 25 = 5 \times 5 \times 5 = 5^3\)
   • Step 2: Group the prime factors into pairs.
   • \( \sqrt{125} = \sqrt{5 \times 5 \times 5} \)
   • Step 3: Take one number from each pair out of the square root.
   • \( \sqrt{125} = 5 \sqrt{5} \)
2. Solution: \( \sqrt{125} = 5\sqrt{5} \)

Problem 2: Simplify the Square Root of 72

1. Prime Factorization Method:
   • Step 1: Find the prime factors of 72.
   • \(72 = 2 \times 36 = 2 \times 6 \times 6 = 2 \times 2 \times 3 \times 3 \times 2 = 2^3 \times 3^2\)
   • Step 2: Group the prime factors into pairs.
   • \( \sqrt{72} = \sqrt{2 \times 2 \times 2 \times 3 \times 3} \)
   • Step 3: Take one number from each pair out of the square root.
   • \( \sqrt{72} = 2 \times 3 \sqrt{2} \)
2. Solution: \( \sqrt{72} = 6\sqrt{2} \)

Problem 3: Simplify the Square Root of 180

1. Prime Factorization Method:
   • Step 1: Find the prime factors of 180.
   • \(180 = 2 \times 90 = 2 \times 2 \times 45 = 2^2 \times 3 \times 15 = 2^2 \times 3 \times 3 \times 5 = 2^2 \times 3^2 \times 5\)
   • Step 2: Group the prime factors into pairs.
   • \( \sqrt{180} = \sqrt{2 \times 2 \times 3 \times 3 \times 5} \)
   • Step 3: Take one number from each pair out of the square root.
   • \( \sqrt{180} = 2 \times 3 \sqrt{5} \)
2. Solution: \( \sqrt{180} = 6\sqrt{5} \)

Problem 4: Simplify the Square Root of 50

1. Prime Factorization Method:
   • Step 1: Find the prime factors of 50.
   • \(50 = 2 \times 25 = 2 \times 5 \times 5 = 2 \times 5^2\)
   • Step 2: Group the prime factors into pairs.
   • \( \sqrt{50} = \sqrt{2 \times 5 \times 5} \)
   • Step 3: Take one number from each pair out of the square root.
   • \( \sqrt{50} = 5 \sqrt{2} \)
2. Solution: \( \sqrt{50} = 5\sqrt{2} \)

Practice Problems

Try simplifying these square roots on your own:

• \(\sqrt{200}\)
• \(\sqrt{288}\)
• \(\sqrt{45}\)
• \(\sqrt{98}\)

Check your answers using the steps outlined above!

Here are some frequently asked questions about simplifying square roots:

• Q: What is the simplified form of the square root of 125?

  A: The square root of 125 simplifies to \(5\sqrt{5}\). This is because 125 can be factored into \(5^2 \times 5\), and the square root of \(5^2\) is 5. Hence, \(\sqrt{125} = \sqrt{5^2 \times 5} = 5\sqrt{5}\).

• Q: What is the prime factorization method for simplifying square roots?

  A: The prime factorization method involves breaking down the number into its prime factors and then simplifying the square root. For example, to simplify \(\sqrt{125}\):
  

  
  
  1. Find the prime factors of 125: \(125 = 5 \times 5 \times 5 = 5^2 \times 5\).
  
  
  2. Rewrite the square root: \(\sqrt{125} = \sqrt{5^2 \times 5}\).
  
  
  3. Simplify by taking the square root of the squared term: \(5\sqrt{5}\).
  
  
  
  


• Q: Can all square roots be simplified using the prime factorization method?

  A: No, not all square roots can be simplified using the prime factorization method. Only square roots of numbers that can be factored into perfect squares can be simplified. For instance, \(\sqrt{12}\) simplifies to \(2\sqrt{3}\) because \(12 = 2^2 \times 3\). However, \(\sqrt{3}\) cannot be simplified further.

• Q: What is an irrational number and how does it relate to square roots?

  A: An irrational number is a number that cannot be expressed as a simple fraction. The square root of a non-perfect square is often an irrational number. For example, \(\sqrt{125} = 5\sqrt{5}\), and \(\sqrt{5}\) is an irrational number, so \(\sqrt{125}\) is also irrational.

• Q: Are there any alternative methods to simplify square roots?

  A: Yes, another method to simplify square roots is the long division method. This method is more complex and involves dividing the number into smaller, more manageable parts, but it is useful for finding the square roots of larger numbers.

Simplifying the square root of 125 involves breaking down the number into its prime factors and then simplifying the expression step-by-step. Here’s a summary of the process:

1. Prime Factorization:

   We start by finding the prime factors of 125. The prime factorization of 125 is \(125 = 5^3\).

2. Group Factors:

   Next, we group the prime factors into pairs of the same number. In this case, we have \(5^2 \times 5\).

3. Extract Square Roots:

   We then take the square root of each group of paired factors. For \(5^2\), the square root is 5. The remaining factor, 5, stays under the radical.

4. Simplify:

   Combining these results, we get the simplified form of the square root of 125: \( \sqrt{125} = 5\sqrt{5} \).

5. Decimal Form:

   For practical applications, it can be useful to know the decimal approximation of \( \sqrt{125} \), which is approximately 11.1803.

By understanding and following these steps, you can simplify the square root of any number using prime factorization. This method not only helps in simplifying square roots but also in gaining a deeper understanding of the number's structure.