63 Square Root Simplified: Easy Step-by-Step Guide

To simplify the square root of 63, we need to find the prime factorization of 63 and look for pairs of factors.

Step-by-Step Instructions:

1. First, find the prime factors of 63.
   • 63 is divisible by 3 (since 6 + 3 = 9, and 9 is divisible by 3).
   • 63 ÷ 3 = 21
   • 21 is divisible by 3 (since 2 + 1 = 3, and 3 is divisible by 3).
   • 21 ÷ 3 = 7
   • 7 is a prime number.
2. So, the prime factorization of 63 is \( 3 \times 3 \times 7 \) or \( 3^2 \times 7 \).
3. Next, rewrite the square root of 63 using its prime factors: \[ \sqrt{63} = \sqrt{3^2 \times 7} \]
4. Separate the square root into the product of square roots: \[ \sqrt{63} = \sqrt{3^2} \times \sqrt{7} \]
5. Simplify the square root of 3 squared: \[ \sqrt{3^2} = 3 \]
6. Therefore: \[ \sqrt{63} = 3 \sqrt{7} \]

So, the simplified form of \( \sqrt{63} \) is \( 3\sqrt{7} \).

Simplifying square roots involves expressing the square root in its simplest radical form. The process typically involves factorizing the number inside the square root and identifying any perfect square factors. Let's explore this process step-by-step using the square root of 63 as an example.

1. List the factors of 63: 1, 3, 7, 9, 21, 63.

2. Identify the perfect squares from the list of factors: 1, 9.

3. Divide 63 by the largest perfect square factor: \( \frac{63}{9} = 7 \).

4. Calculate the square root of the largest perfect square: \( \sqrt{9} = 3 \).

5. Express the original square root as a product of the square root of the perfect square and the square root of the quotient: \( \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7} \).

By following these steps, the square root of 63 simplifies to \( 3\sqrt{7} \). This method can be applied to any square root to simplify it to its most basic form, making calculations and further mathematical operations easier.

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 63 is a number \( q \) such that \( q \times q = 63 \). This is usually represented as \( \sqrt{63} \).

Square roots can be expressed in different forms: the radical form and the decimal form. The radical form is the simplest form of the square root, written with the radical symbol (√). For 63, the simplest radical form is \( \sqrt{63} \). To simplify \( \sqrt{63} \), we can use the factorization method.

Let's go through the step-by-step process to simplify \( \sqrt{63} \):

1. First, list all the factors of 63: 1, 3, 7, 9, 21, 63.
2. Identify the largest perfect square among these factors: in this case, 9.
3. Divide 63 by the largest perfect square: \( 63 \div 9 = 7 \).
4. Take the square root of the largest perfect square: \( \sqrt{9} = 3 \).
5. Combine the results to get the simplest form: \( \sqrt{63} = 3\sqrt{7} \).

Thus, the simplest radical form of the square root of 63 is \( 3\sqrt{7} \), which is approximately equal to 7.937 in decimal form.

Understanding square roots and how to simplify them is fundamental in mathematics as it forms the basis for more complex calculations and problem-solving techniques.

The prime factorization method is a powerful tool for simplifying square roots. This method involves breaking down a number into its prime factors and then simplifying the radical. Let's use this method to simplify the square root of 63.

1. Identify the prime factors of 63. The prime factorization of 63 is \(63 = 3 \times 3 \times 7\), or \(63 = 3^2 \times 7\).
2. Rewrite the square root of 63 using its prime factors: \(\sqrt{63} = \sqrt{3^2 \times 7}\).
3. Apply the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\): \(\sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7}\).
4. Simplify \(\sqrt{3^2}\) to 3: \(\sqrt{3^2} \times \sqrt{7} = 3 \times \sqrt{7}\).
5. Thus, the simplified form of \(\sqrt{63}\) is \(3\sqrt{7}\).

Using the prime factorization method, we have successfully simplified the square root of 63 to \(3\sqrt{7}\). This form is much easier to work with in various mathematical contexts.

The square root of 63 can be simplified using the prime factorization method. Follow these steps for a detailed, step-by-step guide:

1. Find the Prime Factors: Start by finding the prime factors of 63.

   \(63 = 3 \times 3 \times 7\) or \(63 = 3^2 \times 7\)

2. Rewrite the Square Root: Rewrite the square root of 63 using its prime factors.

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

3. Apply the Product Rule for Radicals: Separate the square root into the product of the square roots.

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

4. Simplify: Simplify the square roots of the perfect squares.

   \(\sqrt{3^2} = 3\)

   Therefore, \(\sqrt{63} = 3 \times \sqrt{7}\)

The simplified form of the square root of 63 is \(3\sqrt{7}\).

Understanding how to simplify square roots through examples can make the process clearer. Below are a few example problems along with their solutions to help you master the technique.

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

1. Start by identifying the prime factorization of 63: \( 63 = 3^2 \times 7 \).
2. Apply the square root to the prime factors: \( \sqrt{63} = \sqrt{3^2 \times 7} \).
3. Simplify by taking the square root of the perfect square: \( \sqrt{3^2 \times 7} = 3 \sqrt{7} \).
4. Thus, \( \sqrt{63} = 3 \sqrt{7} \).

Example 2: Determine if \( -\sqrt{63} \) is a valid solution.

1. We know that \( \sqrt{63} = 3\sqrt{7} \approx 7.937 \).
2. Thus, \( -\sqrt{63} = -3\sqrt{7} \approx -7.937 \).
3. This negative value is indeed valid as it represents the negative square root.

Example 3: Calculate the square root of a related problem: \( \sqrt{63 \times 63} \).

1. Express the product inside the square root: \( \sqrt{63 \times 63} \).
2. Simplify the expression: \( \sqrt{63^2} = 63 \).
3. Therefore, \( \sqrt{63 \times 63} = 63 \).

By following these examples, you can practice and become more comfortable with simplifying square roots.

When simplifying square roots, there are several common mistakes that students often make. Being aware of these can help improve accuracy and confidence in solving mathematical problems. Here are some mistakes to watch out for:

• Ignoring Negative Solutions: Remember that the square root of a number can have both positive and negative solutions. For example, \( \sqrt{9} \) is both \( +3 \) and \( -3 \).
• Incorrectly Adding Roots: Square roots can only be added or subtracted when they have the same radicand. Avoid trying to combine \( \sqrt{a} + \sqrt{b} \) unless \( a = b \).
• Misapplying Properties: Ensure that you apply the product and quotient properties correctly. The square root of a product is not the same as the product of the square roots unless applied to each factor individually.
• Overlooking Perfect Squares: When simplifying, always look for perfect squares within the radicand. Missing these can lead to a less simplified form.
• Confusion with Imaginary Numbers: The square root of a negative number involves imaginary numbers. Don't forget that \( \sqrt{-a} = i\sqrt{a} \), where \( i \) is the imaginary unit.
• Calculation Errors: Simple arithmetic mistakes can lead to incorrect simplification. Double-check your factorization and arithmetic operations.

By avoiding these common errors, you can simplify square roots more effectively and accurately, enhancing your mathematical problem-solving skills.

Simplifying square roots, including the square root of 63, has a variety of practical applications in different fields. Understanding how to simplify square roots can help in mathematics, science, engineering, and even everyday tasks. Here are some of the key applications:

• Mathematics:

  Simplifying square roots is a fundamental skill in algebra and higher-level mathematics. It helps in solving equations, simplifying expressions, and understanding geometric concepts. For example, knowing that the square root of 63 simplifies to \(3\sqrt{7}\) can make it easier to solve equations involving \(\sqrt{63}\).

• Geometry:

  In geometry, the simplification of square roots is essential for calculating distances, areas, and volumes. For example, the distance formula in coordinate geometry often involves square roots, and simplifying them can make calculations more straightforward.

• Engineering and Physics:

  Engineers and physicists frequently encounter square roots when dealing with formulas for velocity, acceleration, force, and other physical quantities. Simplifying square roots can lead to more elegant and manageable equations. For instance, simplifying the square root of 63 in a physics problem could streamline the calculation process.

• Computer Science:

  In computer science, algorithms often require the manipulation of mathematical expressions, including square roots. Simplifying these expressions can optimize algorithms and improve computational efficiency.

• Everyday Life:

  Simplifying square roots can also be useful in everyday situations, such as when dealing with measurements, scaling recipes, or working on DIY projects. For example, if you need to cut a piece of wood to a specific diagonal length, simplifying the square root involved in the Pythagorean theorem can help you find the exact measurement more easily.

Overall, simplifying square roots like \(\sqrt{63}\) to \(3\sqrt{7}\) helps in making complex problems more manageable and provides a clearer understanding of the relationships between different mathematical concepts.

Here are some practice problems to help you master the skill of simplifying square roots. Work through each problem and verify your solutions using the step-by-step explanations provided.

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

   Solution:

   1. Identify the prime factors of 63: \(63 = 3^2 \times 7\).
   2. Separate the square root into its factors: \(\sqrt{63} = \sqrt{3^2 \times 7}\).
   3. Pull out the square factor: \(\sqrt{3^2} \times \sqrt{7} = 3\sqrt{7}\).
   4. Final answer: \(3\sqrt{7}\).

2. Simplify \(\sqrt{50}\).

   Solution:

   1. Identify the prime factors of 50: \(50 = 2 \times 5^2\).
   2. Separate the square root into its factors: \(\sqrt{50} = \sqrt{2 \times 5^2}\).
   3. Pull out the square factor: \(\sqrt{5^2} \times \sqrt{2} = 5\sqrt{2}\).
   4. Final answer: \(5\sqrt{2}\).

3. Simplify \(\sqrt{72}\).

   Solution:

   1. Identify the prime factors of 72: \(72 = 2^3 \times 3^2\).
   2. Separate the square root into its factors: \(\sqrt{72} = \sqrt{2^3 \times 3^2}\).
   3. Pull out the square factor: \(\sqrt{2^2 \times 2 \times 3^2} = 2 \times 3 \sqrt{2} = 6\sqrt{2}\).
   4. Final answer: \(6\sqrt{2}\).

Now, try these additional problems on your own:

• Simplify \(\sqrt{45}\).
• Simplify \(\sqrt{18}\).
• Simplify \(\sqrt{98}\).
• Simplify \(\sqrt{200}\).

Check your answers to ensure you are correctly identifying and simplifying the square factors.

For further study and practice on simplifying square roots, including the square root of 63, here are some valuable resources and references:

• Symbolab: A comprehensive tool that offers step-by-step solutions for various mathematical problems, including simplifying square roots.

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• Mathway: This website provides detailed steps on how to simplify the square root of 63 and other mathematical problems. It's a great resource for both students and educators.

  Visit:

• Calculator.name: Offers a simple explanation and a calculator for simplifying square roots, including detailed prime factorization steps.

  Visit:

• Number Maniacs: Provides a step-by-step breakdown of the process to simplify the square root of 63, including identifying factors and perfect squares.

  Visit:

• Cuemath: Features a detailed guide on finding the square root of 63 using the long division method, along with examples and interactive questions.

  Visit:

These resources provide valuable information and practice problems to help you master the concept of simplifying square roots. Explore them to deepen your understanding and improve your skills.

The process of simplifying the square root of 63 showcases the utility and importance of understanding square roots and their applications. By employing methods such as prime factorization and the long division method, we can simplify complex radicals to their simplest forms, which in this case is \( 3\sqrt{7} \). This simplification not only makes the number easier to work with in mathematical problems but also enhances our comprehension of number properties.

Learning to simplify square roots has practical applications across various fields such as engineering, physics, and computer science. For instance, simplifying radicals is crucial in solving quadratic equations, optimizing algorithms, and understanding wave functions in quantum mechanics. It helps in breaking down complex problems into more manageable parts, leading to more efficient and accurate solutions.

Overall, mastering the skill of simplifying square roots, like that of 63, builds a strong foundation for more advanced mathematical concepts and real-world problem-solving. By continuously practicing and applying these methods, students and professionals can enhance their analytical abilities and achieve greater success in their respective fields.