Simplify the Square Root of 63: Easy Step-by-Step Guide

Simplifying square roots involves finding the prime factors of the number and then applying the property of square roots. Here, we will simplify the square root of 63 step-by-step.

Step-by-Step Simplification

1. Find the prime factorization of 63.
2. Rewrite 63 as a product of prime factors.
3. Group the prime factors into pairs.
4. Simplify by taking the square root of the pairs.

Prime Factorization of 63

To find the prime factors of 63, we start with the smallest prime number:

• 63 is divisible by 3: \(63 \div 3 = 21\)
• 21 is divisible by 3: \(21 \div 3 = 7\)
• 7 is a prime number.

So, the prime factorization of 63 is:

\[63 = 3 \times 3 \times 7\]

Grouping Prime Factors

Next, we group the prime factors into pairs:

\[63 = (3 \times 3) \times 7 = 3^2 \times 7\]

Simplifying the Square Root

Now, we apply the property of square roots:

\[\sqrt{63} = \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7}\]

Since the square root of \(3^2\) is 3, we get:

\[\sqrt{63} = 3 \sqrt{7}\]

Final Answer

Therefore, the simplified form of \(\sqrt{63}\) is:

\[\boxed{3 \sqrt{7}}\]

Simplifying square roots is a fundamental skill in mathematics that helps in reducing expressions to their simplest form. The process involves breaking down a number into its prime factors and then applying the properties of square roots. This technique not only makes calculations easier but also helps in understanding the structure of numbers better. Let’s dive into the detailed steps involved in simplifying the square root of 63.

Steps to Simplify the Square Root of 63

1. Find the Prime Factors of the Number

First, identify the prime factors of 63. A prime factor is a factor that is a prime number. For 63, the prime factorization is as follows:

• 63 divided by 3 is 21
• 21 divided by 3 is 7
• 7 is a prime number

So, the prime factors of 63 are: \(63 = 3 \times 3 \times 7\) or \(63 = 3^2 \times 7\).

2. Group the Factors into Pairs

Next, group the prime factors into pairs. This helps in simplifying the square root:

• \(3 \times 3 = 3^2\)

The expression becomes: \(63 = 3^2 \times 7\).

3. Simplify Using Square Root Properties

Apply the square root to each part of the expression. The property of square roots states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\):

• \(\sqrt{63} = \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7}\)
• \(\sqrt{3^2} = 3\)

So, \(\sqrt{63} = 3 \sqrt{7}\).

By following these steps, you can simplify the square root of any number. This method helps in breaking down complex expressions and makes calculations more manageable.

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. To understand prime factorization, let's break it down step-by-step:

1. Identify if the number is prime or composite:

   If the number is prime, it cannot be factored further. If it is composite, it can be broken down into smaller prime factors.

2. Start with the smallest prime number:

   The smallest prime number is 2. Check if the given number is divisible by 2. If it is, divide the number by 2 and continue the process with the quotient. If it is not, move to the next prime number (3, 5, 7, etc.).

3. Repeat the process:

   Continue dividing the number by the smallest possible prime factor until the quotient is a prime number. This quotient is also a part of the prime factors.

Let's apply this process to factorize the number 63:

• 63 is an odd number, so it is not divisible by 2.
• Next, check divisibility by 3. The sum of the digits of 63 (6 + 3) is 9, which is divisible by 3. Therefore, 63 is divisible by 3.
• Divide 63 by 3 to get 21. So, 63 = 3 × 21.
• Now, factorize 21. The sum of the digits of 21 (2 + 1) is 3, which is divisible by 3. Therefore, 21 is divisible by 3.
• Divide 21 by 3 to get 7. So, 21 = 3 × 7.
• Since 7 is a prime number, we cannot factor it further.

Therefore, the prime factorization of 63 is:

63 = 3 × 3 × 7

This breakdown shows how 63 can be expressed as a product of its prime factors, which is essential for simplifying square roots and understanding other mathematical concepts.

To simplify the square root of a number, we need to identify and group pairs of prime factors. Grouping and pairing factors is an essential step in simplifying square roots. Let's go through this process for the number 63.

1. Find the prime factorization of 63:

   From the previous section, we know that the prime factorization of 63 is:

   \[63 = 3 \times 3 \times 7\]

2. Group the factors into pairs:

   We look for pairs of identical factors. In the case of 63, we have a pair of 3s. Grouping the factors, we get:

   \[63 = (3 \times 3) \times 7\]

3. Simplify the square root using the pairs:

   For each pair of identical factors, one factor can be taken out of the square root. Since we have a pair of 3s, we can take 3 out of the square root:

   \[\sqrt{63} = \sqrt{(3 \times 3) \times 7} = 3\sqrt{7}\]

To summarize, the steps involved in grouping and pairing factors for simplifying the square root of 63 are:

• Find the prime factorization of the number.
• Identify and group pairs of identical factors.
• Take one factor out of each pair from under the square root.
• Multiply the factors outside the square root and simplify the expression inside the square root if possible.

Applying these steps to 63, we grouped the factors and simplified the square root to:

\[\sqrt{63} = 3\sqrt{7}\]

By following these steps, we can simplify the square root of any composite number by grouping and pairing its prime factors.

To simplify the square root of a number, we can apply certain properties of square roots. These properties help us break down the expression into simpler components. Let's explore these properties and see how they can be applied to the square root of 63.

1. Property 1: Product of Square Roots

   The square root of a product is equal to the product of the square roots of the factors:

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

   We can use this property to simplify \(\sqrt{63}\) by expressing 63 as a product of its factors.

2. Property 2: Simplifying Perfect Squares

   The square root of a perfect square is a whole number. For example:

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

   When we factorize 63, we look for any perfect square factors.

3. Applying these properties to \(\sqrt{63}\)

   First, we find the prime factorization of 63:

   \[63 = 3 \times 3 \times 7\]

   We can group the factors into a perfect square and another factor:

   \[63 = (3 \times 3) \times 7\]

   Now, apply the product of square roots property:

   \[\sqrt{63} = \sqrt{(3 \times 3) \times 7} = \sqrt{3^2 \times 7}\]

   Next, apply the property of simplifying perfect squares:

   \[\sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} = 3 \times \sqrt{7}\]

Therefore, by applying these square root properties, we have simplified \(\sqrt{63}\) to:

\[\sqrt{63} = 3\sqrt{7}\]

To summarize, the steps for applying square root properties to simplify \(\sqrt{63}\) are:

• Use the product of square roots property to break down the number.
• Identify and simplify any perfect square factors.
• Multiply the simplified square root of the perfect square by the remaining square root.

By following these steps, we can effectively simplify the square root of any composite number.

Simplifying the square root of a number involves breaking down the number into its prime factors and applying square root properties. Here, we will go through the step-by-step simplification process for \(\sqrt{63}\).

1. Step 1: Prime Factorization

   First, find the prime factorization of 63. The prime factorization is the expression of the number as a product of its prime factors:

   \[63 = 3 \times 3 \times 7\]

2. Step 2: Group the Factors

   Next, group the prime factors into pairs. Pairing the factors helps us apply the square root properties effectively:

   \[63 = (3 \times 3) \times 7\]

3. Step 3: Apply the Square Root to Each Group

   Using the property of square roots that states the square root of a product is the product of the square roots, we can separate the factors under the square root:

   \[\sqrt{63} = \sqrt{(3 \times 3) \times 7} = \sqrt{3^2 \times 7}\]

4. Step 4: Simplify the Square Root of the Perfect Square

   The square root of a perfect square is simply the number itself. In our case, the perfect square is \(3^2\):

   \[\sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} = 3 \times \sqrt{7}\]

5. Step 5: Write the Final Simplified Form

   Combine the simplified factors to write the final simplified form of the square root:

   \[\sqrt{63} = 3\sqrt{7}\]

To summarize, the simplified form of \(\sqrt{63}\) is \(3\sqrt{7}\). Here is a table that outlines each step of the simplification process:

By following these steps, you can simplify the square root of any number by breaking it down into its prime factors and applying square root properties.

Let's look at some example problems to better understand the process of simplifying square roots. We'll go through each example step-by-step.

1. Example 1: Simplify \(\sqrt{63}\)

   1. Find the prime factorization of 63:

   \[63 = 3 \times 3 \times 7\]

   2. Group the factors into pairs:

   \[63 = (3 \times 3) \times 7\]

   3. Apply the square root to each group:

   \[\sqrt{63} = \sqrt{(3 \times 3) \times 7} = \sqrt{3^2 \times 7}\]

   4. Simplify the square root of the perfect square:

   \[\sqrt{3^2 \times 7} = 3 \sqrt{7}\]

   5. Write the final simplified form:

   \[\sqrt{63} = 3\sqrt{7}\]

2. Example 2: Simplify \(\sqrt{45}\)

   1. Find the prime factorization of 45:

   \[45 = 3 \times 3 \times 5\]

   2. Group the factors into pairs:

   \[45 = (3 \times 3) \times 5\]

   3. Apply the square root to each group:

   \[\sqrt{45} = \sqrt{(3 \times 3) \times 5} = \sqrt{3^2 \times 5}\]

   4. Simplify the square root of the perfect square:

   \[\sqrt{3^2 \times 5} = 3 \sqrt{5}\]

   5. Write the final simplified form:

   \[\sqrt{45} = 3\sqrt{5}\]

3. Example 3: Simplify \(\sqrt{98}\)

   1. Find the prime factorization of 98:

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

   2. Group the factors into pairs:

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

   3. Apply the square root to each group:

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

   4. Simplify the square root of the perfect square:

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

   5. Write the final simplified form:

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

These examples illustrate the process of simplifying square roots by breaking down the number into its prime factors, grouping pairs, and applying square root properties. By practicing with different numbers, you can become proficient in simplifying square roots.

When simplifying square roots, it's important to be aware of common mistakes that can lead to incorrect results. Here are some common errors to watch out for, along with tips on how to avoid them:

1. Skipping Prime Factorization

   One of the most common mistakes is skipping the prime factorization step and trying to simplify the square root directly. Always start by finding the prime factors of the number.

   Example: For \(\sqrt{63}\), the correct approach is to find the prime factors:

   \[63 = 3 \times 3 \times 7\]

2. Incorrectly Grouping Factors

   Another mistake is incorrectly grouping the factors. Ensure you correctly pair the identical factors.

   Example: For \(\sqrt{63}\), correctly group the factors as:

   \[63 = (3 \times 3) \times 7\]

3. Not Applying Square Root Properties

   Failing to apply square root properties can lead to incorrect simplifications. Use the property that the square root of a product is the product of the square roots.

   Example: For \(\sqrt{63}\), apply the property:

   \[\sqrt{63} = \sqrt{(3 \times 3) \times 7} = \sqrt{3^2 \times 7} = 3\sqrt{7}\]

4. Overlooking Perfect Squares

   Sometimes, perfect squares within the factors are overlooked. Identify and simplify perfect squares correctly.

   Example: In \(\sqrt{63}\), recognize \(3^2\) as a perfect square:

   \[\sqrt{3^2 \times 7} = 3 \sqrt{7}\]

5. Misidentifying Non-Perfect Squares

   Avoid treating non-perfect squares as perfect squares. Ensure the factors you simplify are indeed perfect squares.

   Tip: Remember that only numbers like \(1, 4, 9, 16, 25, \ldots\) are perfect squares.

6. Incorrect Final Simplification

   After applying all steps, ensure the final simplification is correct and there are no errors in multiplication or root extraction.

   Example: The correct simplification for \(\sqrt{63}\) is:

   \[\sqrt{63} = 3\sqrt{7}\]

By being aware of these common mistakes and carefully following the correct steps, you can avoid errors and accurately simplify square roots. Practice regularly and review each step to ensure you have the correct final answer.

Here are some additional practice exercises to help you further understand how to simplify the square root of 63:

1. Simplify the square root of 63 by prime factorization.
2. Practice grouping and pairing factors to simplify the expression.
3. Apply square root properties to simplify expressions involving the square root of 63.
4. Work through step-by-step simplification processes for various expressions containing the square root of 63.
5. Solve example problems involving the square root of 63, ensuring to follow each step carefully.
6. Avoid common mistakes typically made when simplifying square roots.

After exploring various methods to simplify the square root of 63, we have gained a deeper understanding of prime factorization, grouping and pairing factors, and applying square root properties. Through practice exercises and example problems, we've honed our skills in simplifying expressions involving the square root of 63.

In summary, simplifying the square root of 63 involves breaking down the number into its prime factors, grouping and pairing these factors, and applying relevant square root properties. By avoiding common mistakes and diligently following step-by-step processes, we can confidently simplify expressions containing the square root of 63.