How to Simplify the Square Root of 28: A Step-by-Step Guide

The square root of 28 can be simplified by breaking it down into its prime factors and using the properties of square roots.

Step-by-Step Simplification

1. Factorize 28 into its prime factors: \( 28 = 2^2 \times 7 \).
2. Express the square root of 28 using these prime factors: \( \sqrt{28} = \sqrt{2^2 \times 7} \).
3. Separate the square root into the product of square roots: \( \sqrt{2^2} \times \sqrt{7} \).
4. Simplify \( \sqrt{2^2} \) to 2: \( 2 \sqrt{7} \).

Thus, the simplified form of \( \sqrt{28} \) is \( 2 \sqrt{7} \).

Decimal Form

The decimal approximation of \( 2 \sqrt{7} \) is approximately 5.2915.

Properties

• Exact Form: \( 2 \sqrt{7} \)
• Decimal Form: 5.2915026221292
• Rational/Irrational: The square root of 28 is an irrational number.

Additional Examples

Here are some other square roots simplified using the same method:

• Square root of 29: \( \sqrt{29} \approx 5.385 \)
• Square root of 30: \( \sqrt{30} \approx 5.477 \)
• Square root of 31: \( \sqrt{31} \approx 5.568 \)

Understanding how to simplify square roots can help in solving algebraic problems and in other mathematical applications.

FAQs

• What is the square root of 28? The square root of 28 is \( 2 \sqrt{7} \).
• Is the square root of 28 rational? No, it is not rational.
• How do you express the square root of 28 in exponent form? In exponent form, it is written as \( 28^{1/2} \).

Simplifying the square root of 28 involves breaking down the number into its prime factors and applying the properties of square roots. This process makes it easier to work with the square root in various mathematical contexts. By following a few straightforward steps, you can express the square root of 28 in its simplest radical form, which is \(2\sqrt{7}\). This guide will walk you through the methods and provide detailed examples to ensure you master this concept.

The prime factorization method is a straightforward way to simplify the square root of a number by expressing it as a product of its prime factors. Here is a detailed step-by-step process to simplify the square root of 28:

1. Start by factorizing 28 into its prime factors. Divide 28 by the smallest prime number, which is 2:
   • 28 ÷ 2 = 14
   • 14 ÷ 2 = 7
   • 7 is a prime number
   Therefore, the prime factorization of 28 is \( 2^2 \times 7 \).
2. Rewrite the square root of 28 using its prime factors:

   \(\sqrt{28} = \sqrt{2^2 \times 7}\)

3. Apply the property of square roots which states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\):

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

4. Simplify the square root of the perfect square:

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

5. Combine the results to get the simplified form:

   \(\sqrt{28} = 2\sqrt{7}\)

By breaking down 28 into its prime factors and using the properties of square roots, we have simplified \(\sqrt{28}\) to \(2\sqrt{7}\).

To simplify the square root of 28, follow these detailed steps:

1. Factorize 28 into its prime factors: \(28 = 2^2 \times 7\).
2. Rewrite the square root expression: \(\sqrt{28} = \sqrt{2^2 \times 7}\).
3. Use the property of square roots: \(\sqrt{2^2 \times 7} = \sqrt{2^2} \times \sqrt{7}\).
4. Simplify the square root of the perfect square: \(\sqrt{2^2} = 2\).
5. Combine the results: \(\sqrt{28} = 2\sqrt{7}\).

This step-by-step method shows that the square root of 28 simplifies to \(2\sqrt{7}\), making it easier to handle in various mathematical contexts.

When simplified, the square root of 28 is expressed as \(2\sqrt{7}\). To find its decimal form, we calculate the approximate value:

1. First, find the square root of 7, which is approximately 2.6458.
2. Then, multiply this value by 2.
3. Thus, \(2 \times 2.6458 \approx 5.2916\).

So, the square root of 28 in decimal form is approximately 5.2916.

This value is useful when you need a numerical approximation for calculations where an exact radical form isn't practical.

The long division method is a systematic approach to finding the square root of a number without a calculator. It involves a step-by-step division process that is accurate and easy to follow. Here are the detailed steps to simplify the square root of 28 using the long division method:

1. Set up the number: Write 28 as 28.0000 (you can add pairs of zeros for more precision).
2. Group the digits: Pair the digits from right to left, giving us 28 00 00.
3. Find the largest square: Find the largest number whose square is less than or equal to 28. Here, 5 × 5 = 25.
4. Initial division: Divide 28 by 5, giving a quotient of 5 and a remainder of 3. Write 5 above the division bar.
5. Double the quotient: Double the quotient (5), which gives 10. Write it down as the new divisor.
6. Bring down the next pair: Bring down the next pair of zeros to make it 300. Find a digit (x) such that 10x × x is less than or equal to 300. Here, 102 × 2 = 204.
7. Repeat the process: Subtract 204 from 300 to get 96. Bring down the next pair of zeros, making it 9600. Double the current quotient (52) to get 104, and find the next digit (y) such that 104y × y is less than or equal to 9600. Continue this process to get more decimal places as needed.
8. Final result: After repeating the steps, we get a quotient of approximately 5.2915.

Thus, the square root of 28 using the long division method is approximately 5.2915.

Example 1

Simplify the square root of 28 using the prime factorization method.

1. Factorize 28 into its prime factors: \(28 = 2^2 \times 7\).
2. Express the square root of 28 in terms of these factors: \(\sqrt{28} = \sqrt{2^2 \times 7}\).
3. Simplify by taking the square root of the perfect square factor: \(\sqrt{2^2 \times 7} = 2\sqrt{7}\).
4. Thus, the simplified form is \(2\sqrt{7}\).

Example 2

Find the decimal approximation of \(\sqrt{28}\).

1. Using a calculator, find the square root of 28: \(\sqrt{28} \approx 5.2915\).
2. Round to the nearest tenth if necessary: \(5.3\).

Example 3

Using the values of the square roots of 28 and 29, find the square root of 812.

1. Express 812 as a product of 28 and 29: \(812 = 28 \times 29\).
2. Use the property of square roots: \(\sqrt{812} = \sqrt{28 \times 29} = \sqrt{28} \times \sqrt{29}\).
3. Substitute the simplified form of \(\sqrt{28}\): \( \sqrt{28} = 2\sqrt{7} \).
4. Therefore, \( \sqrt{812} = 2\sqrt{7} \times \sqrt{29} = 2\sqrt{203}\).