How Do You Simplify the Square Root of 50: A Simple Guide

Simplifying the square root of 50 involves breaking it down into its prime factors and simplifying it further. Here is a step-by-step guide:

Step-by-Step Solution

1. Find the prime factorization of 50.
2. Pair the prime factors.
3. Simplify the expression.

Detailed Steps

• Prime Factorization of 50

  The prime factors of 50 are 2 and 5, since:

  \( 50 = 2 \times 5^2 \)

• Pairing the Prime Factors

  Identify pairs of prime factors:

  \( \sqrt{50} = \sqrt{2 \times 5^2} \)

• Simplifying the Expression

  Extract the square root of the paired factors:

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

  Therefore, the simplified form of the square root of 50 is:

  \( \sqrt{50} = 5 \sqrt{2} \)

Conclusion

Simplifying square roots involves finding the prime factors, pairing them appropriately, and then extracting the square roots of these pairs. By following the above steps, we find that:

Simplifying the square root of 50 is a fundamental skill in mathematics that helps in understanding more complex algebraic concepts. This process involves breaking down the number under the square root into its prime factors and simplifying further. In this section, we will provide a detailed and step-by-step guide to simplify the square root of 50, making it easier to grasp and apply.

To simplify \( \sqrt{50} \), follow these steps:

1. Identify the prime factors of 50.
2. Group the factors into pairs.
3. Extract the square root of the paired factors.

Let’s break down each of these steps in detail:

• Finding Prime Factors: Determine the prime factors of 50. The prime factors are 2 and 5.
• Grouping Factors: Rewrite the number as a product of its prime factors and pair them: \( 50 = 2 \times 5^2 \).
• Extracting the Square Root: Simplify by taking the square root of the perfect square factor: \( \sqrt{2 \times 5^2} = 5 \sqrt{2} \).

By following these steps, you will have successfully simplified \( \sqrt{50} \) to \( 5 \sqrt{2} \). This method can be applied to any number, providing a clear and straightforward approach to simplifying square roots.

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 x 5 = 25. Square roots are fundamental in various fields such as mathematics, physics, and engineering because they allow us to simplify expressions and solve equations involving squared terms.

The symbol for the square root is √, and the square root of a number 'n' is written as √n. For example, √25 = 5.

Here are some key points to understand square roots:

• The square root of a positive number is always positive.
• Every positive number has two square roots: one positive and one negative. For example, the square roots of 25 are 5 and -5.
• Zero has one square root, which is 0.
• Negative numbers do not have real square roots because no real number squared will result in a negative number. Instead, they have complex square roots.

To simplify square roots, we often use the method of prime factorization. This method involves breaking down a number into its prime factors and then pairing the factors to simplify the square root expression.

Simplifying square roots involves breaking down the number inside the square root into its prime factors and then simplifying it by extracting pairs of factors. Here are the basic steps to simplify square roots:

1. Find the Prime Factorization:

   Determine the prime factors of the number under the square root. Prime factors are the prime numbers that multiply together to give the original number.

2. Group the Prime Factors into Pairs:

   Once you have the prime factors, group them into pairs of the same number. Each pair of prime factors will contribute one factor outside the square root.

3. Simplify by Extracting Pairs:

   For each pair of prime factors, move one of the numbers from inside the square root to outside the square root. If there are any unpaired prime factors, they remain inside the square root.

4. Multiply the Factors Outside the Square Root:

   Multiply all the factors you have moved outside the square root together. This product is the simplified form of the square root.

Here is an example of simplifying the square root of 50 using these steps:

1. Prime Factorization: \(50 = 2 \times 5 \times 5\)

2. Group the Prime Factors: \(50 = 2 \times (5 \times 5)\)

3. Extract Pairs: Since \(5 \times 5\) is a pair, you can take one 5 out of the square root:

   \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\)

4. Final Simplified Form: \(5\sqrt{2}\)

Therefore, the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\).

The prime factorization method is a systematic approach to simplifying square roots. It involves breaking down a number into its prime factors and using those factors to simplify the expression under the square root. Here is a step-by-step guide on how to use the prime factorization method to simplify square roots:

1. Find the Prime Factors: Determine the prime factors of the number under the square root. Prime factors are the smallest prime numbers that multiply together to give the original number.

   For example, to find the prime factors of 50:

   \(50 \div 2 = 25\)

   \(25 \div 5 = 5\)

   \(5 \div 5 = 1\)

   So, the prime factors of 50 are 2 and 5 (with 5 repeated twice): \(50 = 2 \times 5 \times 5\).

2. Group the Prime Factors into Pairs: Arrange the prime factors into pairs of identical numbers. Each pair of prime factors will simplify to a single number outside the square root.

   For 50, we can group the factors as: \(50 = 2 \times (5 \times 5)\).

3. Move Each Pair Outside the Square Root: For every pair of prime factors, move one factor outside the square root.

   In this case, the pair of 5s can be moved outside the square root:

   \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\).

4. Simplified Form: Combine the numbers outside the square root to get the simplified form of the original square root.

   So, the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\).

Using the prime factorization method helps to systematically simplify square roots, making it easier to handle and solve mathematical expressions.

Simplifying the square root of a number involves expressing it in its simplest radical form. Let's simplify the square root of 50 step by step.

Step-by-Step Guide

1. First, find the prime factorization of 50.
2. Pair the prime factors.
3. Extract the square root of the paired factors.
4. Write the simplified form.

Finding Prime Factors

Prime factorization involves breaking down the number into its prime factors. For 50, we can write:

\[ 50 = 2 \times 25 \]

Notice that 25 is not a prime number, so we further factorize 25:

\[ 25 = 5 \times 5 \]

Thus, the prime factorization of 50 is:

\[ 50 = 2 \times 5 \times 5 \]

Pairing the Prime Factors

In the prime factorization of 50, we look for pairs of the same number:

\[ 50 = 2 \times (5 \times 5) \]

Here, we have one pair of 5's.

Extracting the Square Root

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

\[ \sqrt{50} = \sqrt{2 \times 5^2} \]

The square root of \( 5^2 \) is 5:

\[ \sqrt{50} = 5 \sqrt{2} \]

Simplified Form

The simplified form of the square root of 50 is:

\[ \sqrt{50} = 5 \sqrt{2} \]

This is the simplest form, as \( \sqrt{2} \) cannot be simplified further.

Visual Representation

By following these steps, you can simplify the square root of any number using prime factorization.

Simplifying the square root of 50 involves a series of systematic steps. Follow this detailed guide to achieve the simplest form of the square root of 50.

1. Find the Prime Factorization of 50

   Prime factorization is the process of breaking down a number into its basic prime number factors. Start with the smallest prime number that divides 50.

   \[ 50 \div 2 = 25 \]

   Next, factorize 25, which is not a prime number.

   \[ 25 \div 5 = 5 \]

   Since 5 is a prime number, the prime factors of 50 are:

   \[ 50 = 2 \times 5 \times 5 \]

2. Group the Prime Factors

   Identify pairs of the same prime numbers within the factorization:

   \[ 50 = 2 \times (5 \times 5) \]

   Here, \( 5 \times 5 \) forms a pair.

3. Extract the Square Root of the Pairs

   For each pair of prime factors, take one factor out of the square root. The remaining factors stay inside the square root:

   \[ \sqrt{50} = \sqrt{2 \times 5^2} \]

   The square root of \( 5^2 \) is 5, so we extract it out of the square root:

   \[ \sqrt{50} = 5 \sqrt{2} \]

4. Write the Simplified Form

   The final simplified form of the square root of 50 is:

   \[ \sqrt{50} = 5 \sqrt{2} \]

This step-by-step guide helps you simplify any square root by using prime factorization. Remember, the key is to identify pairs of prime factors and extract them from under the square root.

To simplify the square root of 50, the first step is to find its prime factors. Prime factorization breaks down a number into the product of prime numbers. Follow these steps to determine the prime factors of 50:

1. Start with the smallest prime number

   Begin by dividing 50 by the smallest prime number, which is 2. Check if 50 is divisible by 2:

   \[ 50 \div 2 = 25 \]

   Since 50 is divisible by 2, write down 2 as one of the prime factors and use 25 for the next step.

2. Factorize the quotient

   Next, take the quotient (25) and find its prime factors. Start with the smallest prime number again:

   \[ 25 \div 2 = 12.5 \]

   Since 25 is not divisible by 2, move to the next smallest prime number, which is 3:

   \[ 25 \div 3 = 8.33 \]

   25 is not divisible by 3. The next smallest prime number is 5:

   \[ 25 \div 5 = 5 \]

   Since 25 is divisible by 5, write down 5 as another prime factor. Repeat the process with the quotient:

   \[ 5 \div 5 = 1 \]

   Now that we have reached 1, the factorization process is complete.

3. List all the prime factors

   Combine all the prime factors identified in the steps above:

   \[ 50 = 2 \times 5 \times 5 \]

   Thus, the prime factorization of 50 is:

   \[ 50 = 2 \times 5^2 \]

By following these steps, you can find the prime factors of any number. For 50, the prime factors are 2 and 5.

Once you have the prime factors of 50, the next step is to pair these factors to simplify the square root. Here's how you can do it:

1. Identify the prime factors

   From the previous step, we found the prime factors of 50:

   \[ 50 = 2 \times 5 \times 5 \]

2. Group the prime factors into pairs

   Look for pairs of the same prime factors. In this case, we have a pair of 5s:

   \[ 50 = 2 \times (5 \times 5) \]

   This can be written to highlight the pair:

   \[ 50 = 2 \times 5^2 \]

3. Prepare to simplify

   By pairing the prime factors, we set the stage for extracting them from under the square root. Remember, a pair of the same number under the square root can be simplified to a single number outside the square root:

   \[ \sqrt{50} = \sqrt{2 \times 5^2} \]

This pairing process helps to simplify the expression under the square root, making it easier to extract the square root of the pairs.

After pairing the prime factors, the next step is to extract the square root. This involves simplifying the expression under the square root by taking the square root of each pair of prime factors. Follow these detailed steps:

1. Write the expression with paired prime factors

   From the previous step, we have the paired prime factors of 50:

   \[ \sqrt{50} = \sqrt{2 \times 5^2} \]

2. Extract the square root of the paired factors

   For each pair of prime factors under the square root, take one factor out of the square root. The square root of \( 5^2 \) is 5:

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

   This simplifies the square root expression by moving the 5 outside the square root:

   \[ \sqrt{50} = 5 \sqrt{2} \]

3. Simplify the remaining expression

   The remaining factor under the square root (2) cannot be simplified further, so the expression stays as it is:

   \[ 5 \sqrt{2} \]

The final simplified form of the square root of 50 is:

\[ \sqrt{50} = 5 \sqrt{2} \]

This process of extracting the square root by simplifying the pairs helps in achieving the simplest form of the square root expression.

After extracting the square root of the paired prime factors, we arrive at the simplified form. Here’s a detailed explanation of how to write and verify the simplified form of the square root of 50:

1. Combine the simplified factors

   From the previous steps, we have extracted the factor 5 from the square root, leaving the remaining factor 2 inside the square root:

   \[ \sqrt{50} = 5 \sqrt{2} \]

2. Understand the simplified form

   The expression \( 5 \sqrt{2} \) means that 5 is outside the square root and the square root of 2 remains under the radical sign. This is the simplest form because the square root of 2 cannot be simplified further.

3. Verify the simplification

   To verify, you can compare the decimal approximations of both the original square root and the simplified form:

   • Calculate the square root of 50 directly:

   \[ \sqrt{50} \approx 7.071 \]

   • Calculate the simplified form:

   \[ 5 \times \sqrt{2} \approx 5 \times 1.414 = 7.071 \]

   Both calculations yield approximately the same value, confirming that \( 5 \sqrt{2} \) is indeed the simplified form of \( \sqrt{50} \).

Thus, the simplest radical form of the square root of 50 is:

\[ \sqrt{50} = 5 \sqrt{2} \]

By following these steps, you can ensure the accuracy and simplicity of your square root expressions.

Visualizing the process of simplifying the square root of 50 can help in understanding each step more clearly. Here is a detailed visual representation of the steps involved:

This table summarizes the steps and shows the mathematical expressions at each stage, providing a clear visual guide to simplifying the square root of 50.

To further understand the process of simplifying square roots, let's look at some additional examples. Follow the step-by-step method for each example to simplify the square roots.

Example 1: Simplifying \(\sqrt{72}\)

1. Find the Prime Factorization

   First, break down 72 into its prime factors:

   \[ 72 = 2 \times 36 \]

   Continue factorizing 36:

   \[ 36 = 2 \times 18 \]

   Further factorize 18:

   \[ 18 = 2 \times 9 \]

   And factorize 9:

   \[ 9 = 3 \times 3 \]

   Thus, the prime factorization of 72 is:

   \[ 72 = 2 \times 2 \times 2 \times 3 \times 3 \]

2. Pair the Prime Factors

   Group the prime factors into pairs:

   \[ 72 = (2 \times 2) \times 2 \times (3 \times 3) \]

   Or more clearly:

   \[ 72 = 2^3 \times 3^2 \]

3. Extract the Square Root

   Take the square root of each pair of prime factors:

   \[ \sqrt{72} = \sqrt{2^2 \times 2 \times 3^2} \]

   Extracting the pairs:

   \[ \sqrt{72} = 2 \times 3 \sqrt{2} \]

   So the simplified form is:

   \[ \sqrt{72} = 6 \sqrt{2} \]

Example 2: Simplifying \(\sqrt{98}\)

1. Find the Prime Factorization

   Break down 98 into its prime factors:

   \[ 98 = 2 \times 49 \]

   Further factorize 49:

   \[ 49 = 7 \times 7 \]

   So the prime factorization of 98 is:

   \[ 98 = 2 \times 7 \times 7 \]

2. Pair the Prime Factors

   Group the prime factors into pairs:

   \[ 98 = 2 \times (7 \times 7) \]

   Or more clearly:

   \[ 98 = 2 \times 7^2 \]

3. Extract the Square Root

   Take the square root of each pair of prime factors:

   \[ \sqrt{98} = \sqrt{2 \times 7^2} \]

   Extracting the pairs:

   \[ \sqrt{98} = 7 \sqrt{2} \]

   So the simplified form is:

   \[ \sqrt{98} = 7 \sqrt{2} \]

Example 3: Simplifying \(\sqrt{200}\)

1. Find the Prime Factorization

   Break down 200 into its prime factors:

   \[ 200 = 2 \times 100 \]

   Continue factorizing 100:

   \[ 100 = 2 \times 50 \]

   Further factorize 50:

   \[ 50 = 2 \times 25 \]

   And factorize 25:

   \[ 25 = 5 \times 5 \]

   So the prime factorization of 200 is:

   \[ 200 = 2 \times 2 \times 2 \times 5 \times 5 \]

2. Pair the Prime Factors

   Group the prime factors into pairs:

   \[ 200 = (2 \times 2) \times 2 \times (5 \times 5) \]

   Or more clearly:

   \[ 200 = 2^3 \times 5^2 \]

3. Extract the Square Root

   Take the square root of each pair of prime factors:

   \[ \sqrt{200} = \sqrt{2^2 \times 2 \times 5^2} \]

   Extracting the pairs:

   \[ \sqrt{200} = 2 \times 5 \sqrt{2} \]

   So the simplified form is:

   \[ \sqrt{200} = 10 \sqrt{2} \]

These examples illustrate how to simplify the square roots of different numbers by following the same method of prime factorization, pairing the factors, and extracting the square root.

Here are some practice problems to help you get better at simplifying square roots. Try to follow the steps outlined earlier to find the simplified forms of these square roots.

1. Simplify the square root of 75.
   1. Find the prime factors of 75.
   2. Pair the prime factors.
   3. Extract the square root.
   4. Write the simplified form.
2. Simplify the square root of 98.
   1. Find the prime factors of 98.
   2. Pair the prime factors.
   3. Extract the square root.
   4. Write the simplified form.
3. Simplify the square root of 45.
   1. Find the prime factors of 45.
   2. Pair the prime factors.
   3. Extract the square root.
   4. Write the simplified form.
4. Simplify the square root of 200.
   1. Find the prime factors of 200.
   2. Pair the prime factors.
   3. Extract the square root.
   4. Write the simplified form.

For additional practice, try simplifying the square roots of the following numbers:

• \(\sqrt{32}\)
• \(\sqrt{54}\)
• \(\sqrt{72}\)
• \(\sqrt{108}\)
• \(\sqrt{125}\)

Check your answers using the methods and steps discussed in this guide. Remember, practice makes perfect!

When simplifying square roots, it's easy to make mistakes. Here are some common errors to watch out for and tips to avoid them:

1. Not Fully Factoring the Number:

   Ensure you factor the number completely into its prime factors. Missing a factor can lead to incorrect simplification.

   Example: For \(\sqrt{50}\), factorize as \(50 = 2 \times 5 \times 5\), not \(50 = 25 \times 2\) and stop there.

2. Incorrectly Pairing Factors:

   Remember to pair the prime factors correctly and only once.

   Example: For \(\sqrt{50}\), pair the two 5s: \(50 = 2 \times 5 \times 5\) leads to \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\).

3. Leaving Square Root Unchanged:

   Sometimes, people leave the square root in its original form without simplification.

   Example: Instead of leaving \(\sqrt{50}\) as it is, simplify it to \(5\sqrt{2}\).

4. Incorrect Simplification of Non-Perfect Squares:

   Ensure that you only simplify the square root of perfect square factors.

   Example: For \(\sqrt{72}\), factorize as \(72 = 2^3 \times 3^2\), so \(\sqrt{72} = \sqrt{2^3 \times 3^2} = 3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}\).

5. Forgetting to Simplify Further:

   After initial simplification, always check if further simplification is possible.

   Example: For \(\sqrt{18}\), initial simplification gives \(\sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2}\).

By avoiding these common mistakes, you can ensure your simplifications are accurate and efficient.

• What is the square root of 50?

  The square root of 50 is approximately \( \sqrt{50} = 7.071 \). In simplified radical form, it is expressed as \( 5\sqrt{2} \).

• What is the square of 50?

  The square of 50 is \( 50^2 = 2500 \).

• Is the square root of 50 a rational number?

  No, the square root of 50 is an irrational number. It cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating.

• What is the exponent form of the square root of 50?

  The exponent form of the square root of 50 is \( 50^{1/2} \).

• How can I simplify the square root of 50?

  To simplify the square root of 50, you can use prime factorization:
  

  
  
  1. Factor 50 into its prime factors: \( 50 = 2 \times 5^2 \).
  
  
  2. Express the square root: \( \sqrt{50} = \sqrt{2 \times 5^2} \).
  
  
  3. Simplify by taking the square root of the perfect square: \( \sqrt{50} = 5\sqrt{2} \).
  
  
  
  



• Why is the square root of 50 considered an irrational number?

  Because its decimal representation, 7.071067811865475, goes on forever without repeating, it cannot be expressed as a fraction of two integers.

For a deeper understanding of how to simplify square roots and to explore more examples, consider the following resources: