Simplify Square Root of 63 Easily

To simplify the square root of 63, we follow these steps:

Steps to Simplify

1. Factor 63 into its prime factors: \( 63 = 3^2 \times 7 \).
2. Write the square root of 63 using these factors: \( \sqrt{63} = \sqrt{3^2 \times 7} \).
3. Apply the product rule for radicals: \( \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} \).
4. Simplify the square root of the perfect square: \( \sqrt{3^2} = 3 \).
5. Combine the results: \( \sqrt{63} = 3 \sqrt{7} \).

Result

The simplest radical form of the square root of 63 is:

$$

63

=
3

7

Decimal Form

The square root of 63 in decimal form is approximately:

7.93725

Exponent Form

Expressed with exponents, the square root of 63 is:

\( 63^{1/2} = 3 \times 7^{1/2} \)

Simplifying the square root of 63 involves breaking down the number into its prime factors and identifying any perfect squares. The number 63 can be factored into 3² × 7, which allows us to simplify √63 to 3√7. This process helps in expressing the square root in its simplest radical form, making it easier to work with in various mathematical contexts.

1. Identify the factors of 63: 1, 3, 7, 9, 21, 63
2. Find the perfect squares among these factors: 1, 9
3. Divide 63 by the largest perfect square: 63 / 9 = 7
4. Take the square root of the largest perfect square: √9 = 3
5. Combine the results: 3√7

Using this method, we see that √63 simplifies to 3√7, which is approximately 7.937 in decimal form. This simplified form is particularly useful for further mathematical operations and applications.

The process of simplifying the square root of 63 involves understanding basic mathematical principles, including prime factorization and the properties of square roots. Simplification allows us to express a square root in its simplest form, making it easier to work with in mathematical calculations.

Let's explore the step-by-step process to simplify the square root of 63:

1. Identify the prime factors of 63: The prime factors are the prime numbers that multiply together to give 63. We can factorize 63 as follows:
   • 63 ÷ 3 = 21
   • 21 ÷ 3 = 7
   • Thus, the prime factors of 63 are 3 × 3 × 7 or \(3^2 \times 7\).
2. Recognize the perfect square within the factors: Among the prime factors, \(3^2\) is a perfect square.
3. Apply the square root to the perfect square: We take the square root of \(3^2\), which is 3. This leaves the expression under the square root simplified:
   • \(\sqrt{63} = \sqrt{3^2 \times 7} = 3 \sqrt{7}\).

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

This simplification is beneficial because it makes further calculations more straightforward and helps in understanding the properties and relationships of numbers. Understanding these basic concepts is essential for solving more complex mathematical problems involving square roots.

The prime factorization method is an efficient way to simplify the square root of a number by breaking it down into its prime factors. Here, we will apply this method to simplify the square root of 63 step-by-step.

1. Prime Factorization of 63:

   First, find the prime factors of 63. The prime factors of 63 are 3 and 7. Therefore, we can express 63 as:

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

2. Rewrite the Square Root:

   Rewrite the square root of 63 using its prime factorization:

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

3. Separate the Radicals:

   Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the factors under the square root:

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

4. Simplify the Expression:

   Simplify \(\sqrt{3^2}\) to 3, as the square root and the square cancel each other out:

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

5. Final Simplified Form:

   The simplified form of the square root of 63 is:

   \(\sqrt{63} = 3\sqrt{7}\)

Thus, by using the prime factorization method, we have simplified the square root of 63 to \(3\sqrt{7}\).

The long division method is a systematic approach to finding the square root of a number. Follow these steps to simplify the square root of 63 using this method:

1. Step 1: Pair the digits of the number from right to left. For 63, we treat it as a single pair since it has only two digits.
2. Step 2: Find the largest number whose square is less than or equal to 63. Here, 7 × 7 = 49 is the largest perfect square.
3. Step 3: Subtract this square from 63 (63 - 49 = 14). Bring down pairs of zeros to continue the process. The next pair is 1400.
4. Step 4: Double the quotient obtained so far (which is 7), giving us 14. Now, find a number x such that 14x × x is less than or equal to 1400. Here, x = 9 (since 149 × 9 = 1341).
5. Step 5: Subtract 1341 from 1400, bringing down the next pair of zeros. The result is 59,000.
6. Step 6: Continue this process, bringing down pairs of zeros and finding appropriate divisors. Double the current quotient (which is now 79) and find a digit x such that 158x × x is less than or equal to the new number.

By following these steps, you can continue to find more decimal places for the square root of 63, which approximates to 7.937.

The square root of 63 can be expressed in both exact and decimal forms. Understanding these representations helps in various mathematical applications.

• Exact Form: To express the square root of 63 in its simplest radical form, we use the prime factorization method. The number 63 can be factorized as 3 × 3 × 7, which simplifies to 3√7. Therefore, the exact form of √63 is 3√7.
• Decimal Form: When calculating the decimal form of the square root of 63, we find it to be approximately 7.937. This value is obtained using methods like long division or a calculator and is useful for practical calculations where an approximate value is sufficient.

In summary, the square root of 63 is represented exactly as 3√7 and approximately as 7.937 in decimal form. These forms provide both precise and practical ways to work with this irrational number.

Practicing how to simplify the square root of 63 will help solidify your understanding. Below are step-by-step examples and practice problems to guide you through the process.

1. Example 1: Simplify √63
   • Step 1: List the factors of 63: 1, 3, 7, 9, 21, 63
   • Step 2: Identify the largest perfect square factor: 9
   • Step 3: Express 63 as a product of its factors: 63 = 9 × 7
   • Step 4: Simplify using the square root property: √63 = √(9 × 7) = √9 × √7 = 3√7
2. Example 2: Convert √63 to its decimal form
   • Use a calculator to find the approximate value of √63
   • Result: √63 ≈ 7.937
3. Practice Problem 1: Simplify √45
   • List the factors, identify the largest perfect square, and simplify
4. Practice Problem 2: Convert √75 to its simplest radical form and decimal form
   • Follow the steps outlined in the examples to solve
5. Practice Problem 3: Simplify √20
   • Work through the factorization and simplification process

• Is the square root of 63 rational?
• What is the square number of 63?
• How can I simplify the square root of 63?
• Can I use a calculator to find the square root of 63?
• Are there any special properties of the square root of 63?

After exploring various methods to simplify the square root of 63, it's evident that this task can be accomplished using different approaches. Whether through prime factorization or long division, the process involves breaking down the number into its factors and identifying perfect squares. Additionally, understanding the exact and decimal forms of the square root of 63 provides flexibility in mathematical computations.