Square Root of 48 Simplified: Easy Step-by-Step Guide

The process of simplifying the square root of 48 involves expressing it in its simplest radical form. Let's break down the steps:

Step-by-Step Simplification

1. Factorize 48 into its prime factors:

   48 = 2 × 2 × 2 × 2 × 3

2. Group the factors into pairs of the same number:

   (2 × 2) and (2 × 2) and 3

3. Take the square root of each pair and multiply:

   √(2 × 2) = 2 and √(2 × 2) = 2

4. Multiply these results together and include any remaining factors under the square root:

   2 × 2 × √3 = 4√3

Therefore, the simplified form of √48 is 4√3.

Decimal and Exponent Forms

• The decimal form of the square root of 48 is approximately 6.928.
• In exponent form, it is written as \(48^{1/2} = 4 \cdot 3^{1/2}\).

Conclusion

Simplifying the square root of 48 to its radical form 4√3 allows for easier manipulation in mathematical operations and a clearer understanding of its value. This process involves prime factorization, grouping factors, and taking square roots of pairs.

The square root of a number is a value that, when multiplied by itself, gives the original number. In this guide, we will explore the methods to simplify the square root of 48, a process that involves breaking down the number into its prime factors and using mathematical rules to express it in a simpler form.

Simplifying square roots is a useful skill in algebra that can make complex calculations more manageable. The simplified form of the square root of 48 provides a clearer understanding of its exact value and helps in various mathematical applications.

We will cover several methods to achieve this, including:

• Prime Factorization Method: This involves expressing the number as a product of its prime factors.
• Long Division Method: A step-by-step approach to finding the square root by division.
• Perfect Squares Method: Identifying perfect square factors of the number.

By the end of this guide, you will be able to simplify the square root of 48 and understand its representation in different forms such as decimal and exponent form.

Square roots are fundamental mathematical concepts that are widely used in various fields such as algebra, geometry, and physics. Understanding square roots involves recognizing that the square root of a number \( a \) is a value \( x \) such that \( x^2 = a \). This can be represented mathematically as:

\[ \sqrt{a} = x \quad \text{if and only if} \quad x^2 = a \]

Here are some key points to understand about square roots:

• Radical Symbol: The square root of a number is represented by the radical symbol \( \sqrt{} \). For example, the square root of 48 is written as \( \sqrt{48} \).
• Perfect Squares: Numbers like 1, 4, 9, 16, and 25 are perfect squares because they can be expressed as the square of an integer (e.g., \( 1 = 1^2 \), \( 4 = 2^2 \), \( 9 = 3^2 \)).
• Irrational Numbers: Not all square roots are whole numbers. When the square root of a number is not an integer and cannot be expressed as a simple fraction, it is called an irrational number. For instance, \( \sqrt{2} \) and \( \sqrt{48} \) are irrational.
• Prime Factorization: Simplifying square roots often involves breaking down the number into its prime factors. For example, 48 can be factored into \( 2 \times 2 \times 2 \times 2 \times 3 \), which helps in simplifying \( \sqrt{48} \) to \( 4\sqrt{3} \).

Let's delve deeper into how these principles apply to simplifying the square root of 48 and other numbers in the subsequent sections.

Simplifying the square root of 48 can be approached using several methods. Here, we will discuss three primary methods: Prime Factorization, Long Division, and using Perfect Squares.

1. Prime Factorization Method

Prime factorization involves breaking down the number 48 into its prime factors. Here's a step-by-step guide:

1. Find the prime factors of 48:

   48 = 2 × 2 × 2 × 2 × 3

2. Group the factors in pairs:

   (2 × 2) and (2 × 2) and 3

3. Take one factor from each pair and multiply:

   2 × 2 = 4

4. Combine the result with the remaining factor under the square root:

   \(\sqrt{48} = \sqrt{(2 \times 2) \times (2 \times 2) \times 3} = 4\sqrt{3}\)

2. Long Division Method

The long division method is another approach to find the square root of 48. Follow these steps:

1. Pair up the digits of 48 from right to left.
2. Find a number whose square is less than or equal to 48. In this case, 6 (since \(6^2 = 36\)).
3. Subtract the square of this number from 48:

   48 - 36 = 12

4. Double the divisor and guess the next digit in the quotient. Repeat the process to get a more accurate result.
5. The quotient obtained gives the square root in decimal form:

   \(\sqrt{48} \approx 6.928\)

3. Perfect Squares Method

This method involves using known perfect squares to simplify the square root. Here are the steps:

1. List the factors of 48:

   1, 2, 3, 4, 6, 8, 12, 16, 24, 48

2. Identify the largest perfect square factor:

   16 is the largest perfect square factor of 48.

3. Divide 48 by this perfect square:

   48 / 16 = 3

4. Simplify using the square root of the perfect square:

   \(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)

Each of these methods leads to the simplified form of the square root of 48, which is \(4\sqrt{3}\). Choosing a method depends on your preference and the context in which you are working.

The prime factorization method is a systematic way of simplifying square roots by breaking down the number into its prime factors. Here are the steps to simplify the square root of 48 using prime factorization:

1. First, find the prime factors of 48. We start by dividing 48 by the smallest prime number, which is 2:

   \(48 \div 2 = 24\)

   Then we continue factoring 24:

   \(24 \div 2 = 12\)

   Continuing this process:

   \(12 \div 2 = 6\)

   \(6 \div 2 = 3\)

   Now, we have the prime factorization of 48:

   \(48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3\)

2. Group the prime factors into pairs. Since we are looking for the square root, each pair of prime factors will come out of the square root:

   \(\sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{(2^2)^2 \times 3} = \sqrt{(2^2)} \times \sqrt{(2^2)} \times \sqrt{3}\)

3. Simplify the pairs. Each pair of \(2^2\) becomes 2 when taken out of the square root:

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

   So, we have:

   \(\sqrt{48} = 2 \times 2 \times \sqrt{3} = 4\sqrt{3}\)

4. Thus, the simplified form of the square root of 48 is:

   \(\sqrt{48} = 4\sqrt{3}\)

Using the prime factorization method, we have shown that the square root of 48 simplifies to \(4\sqrt{3}\), making it easier to work with in various mathematical contexts.

The long division method is a systematic approach to find the square root of a number, including non-perfect squares. Below are the detailed steps to find the square root of 48 using the long division method:

1. Step 1: Start by writing the number 48 as 48.000000 to allow for decimal places. Pair the digits from right to left, placing a bar over each pair. So, we have 48.

2. Step 2: Find the largest number whose square is less than or equal to 48. In this case, 6 * 6 = 36, which is less than 48. Write 6 as the quotient and subtract 36 from 48 to get a remainder of 12.

3. Step 3: Double the quotient (6), getting 12, and bring down the next pair of zeros, making the new dividend 1200. Now, find a digit (D) such that (120 + D) * D is less than or equal to 1200. The digit 9 works here, as (120 + 9) * 9 = 1161. Write 9 next to the quotient and subtract 1161 from 1200 to get a remainder of 39.

4. Step 4: Bring down the next pair of zeros, making the new dividend 3900. Double the current quotient (69), making it 138, and find a digit (D) such that (1380 + D) * D is less than or equal to 3900. The digit 2 works here, as (1380 + 2) * 2 = 2764. Write 2 next to the quotient and subtract 2764 from 3900 to get a remainder of 1136.

5. Step 5: Continue this process until you achieve the desired precision. The approximate value of the square root of 48, rounded to three decimal places, is 6.928.

This method helps in finding the square root of any number by breaking it down step-by-step, ensuring precision and understanding of the underlying process.

The perfect squares method is a straightforward approach to simplify the square root of a number by utilizing the largest perfect square factor of that number. Here's a step-by-step guide on how to simplify the square root of 48 using this method:

1. Identify the Factors of 48:
   • The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
2. Find the Perfect Squares:
   • From the list of factors, identify the perfect squares: 1, 4, 16.
3. Choose the Largest Perfect Square:
   • The largest perfect square factor of 48 is 16.
4. Divide 48 by the Largest Perfect Square:
   • \( \frac{48}{16} = 3 \)
5. Calculate the Square Root:
   • \( \sqrt{16} = 4 \)
6. Express the Simplified Form:
   • Combine the results from steps 4 and 5 to get the simplified form:
   • \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \)

Thus, the square root of 48 in its simplest radical form is \( 4\sqrt{3} \).

To calculate the square root of 48 and simplify it, we can follow a few straightforward steps. Here's a detailed method to achieve the result:

1. Factor the number 48 into its prime factors:

   \[ 48 = 2 \times 2 \times 2 \times 2 \times 3 \]

2. Group the factors into pairs of two:

   \[ 48 = (2 \times 2) \times (2 \times 2) \times 3 = 4 \times 4 \times 3 \]

3. Rewrite the expression under the square root sign, noting that the square root of a product is the product of the square roots:

   \[ \sqrt{48} = \sqrt{4 \times 4 \times 3} \]

4. Extract the square roots of the perfect squares (4 in this case):

   \[ \sqrt{4 \times 4 \times 3} = \sqrt{4} \times \sqrt{4} \times \sqrt{3} = 2 \times 2 \times \sqrt{3} \]

5. Simplify the expression further:

   \[ 2 \times 2 \times \sqrt{3} = 4\sqrt{3} \]

6. Thus, the simplified form of the square root of 48 is:

   \[ \sqrt{48} = 4\sqrt{3} \]

7. For the decimal form, we calculate the numerical value of \( \sqrt{48} \) using a calculator:

   \[ \sqrt{48} \approx 6.928 \]

In summary, the simplified radical form of the square root of 48 is \( 4\sqrt{3} \), and its approximate decimal form is 6.928.

The square root of 48, when expressed in its decimal form, is approximately 6.9282032302755. This value is obtained by using the long division method or a calculator to find the most precise decimal representation.

To arrive at this value, you can follow these steps:

1. Identify the two perfect squares that 48 lies between. These are 36 (6²) and 49 (7²).
2. Since 48 is closer to 49 than to 36, we know that the square root of 48 will be closer to 7 than to 6.
3. Using the long division method or a calculator, we can determine the more precise decimal value of the square root of 48, which is approximately 6.9282032302755.

Here is the step-by-step long division method to find the decimal form:

1. Start by grouping the digits in pairs from the decimal point. Since 48 has only two digits, we consider it as 48.000000.
2. Find the largest number whose square is less than or equal to 48. The number is 6 because 6² = 36.
3. Subtract 36 from 48 to get a remainder of 12. Bring down a pair of zeros to make it 1200.
4. Double the quotient (which is 6) to get 12, and use it as the first digit of the new divisor. Find the digit X such that 12X multiplied by X is less than or equal to 1200. The digit is 9, as 129 x 9 = 1161.
5. Subtract 1161 from 1200 to get 39, and bring down another pair of zeros to make it 3900. Continue this process to obtain more decimal places.

Using these steps, you can verify that the square root of 48, in its decimal form, is approximately 6.9282032302755.

Understanding the exponent form of the simplified square root of 48 can help in various mathematical calculations and expressions. Here, we will convert the simplified square root into its exponent form.

First, let’s recall the prime factorization of 48:

• 48 = 2 × 24
• 24 = 2 × 12
• 12 = 2 × 6
• 6 = 2 × 3

So, the prime factorization of 48 is:

48 = 2⁴ × 3

Now, we can find the square root of 48:

\(\sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{2^4} \times \sqrt{3}\)

We know that \(\sqrt{2^4} = 2^2\), so:

\(\sqrt{48} = 2^2 \times \sqrt{3} = 4\sqrt{3}\)

To express this in exponent form:

Remember that any number to the power of \(\frac{1}{2}\) represents the square root of that number.

\(\sqrt{48}\) can be written as:

\(48^{\frac{1}{2}}\)

Using the prime factorization, we get:

\( (2^4 \times 3)^{\frac{1}{2}} \)

Apply the exponent rule \((a \times b)^n = a^n \times b^n\):

\( 2^{4 \times \frac{1}{2}} \times 3^{\frac{1}{2}} \)

This simplifies to:

\( 2^2 \times 3^{\frac{1}{2}} \)

Therefore, the exponent form of the simplified square root of 48 is:

\( 4 \times 3^{\frac{1}{2}} \)

When simplifying the square root of 48, it's important to avoid common mistakes. Here are some key points to keep in mind:

• Incorrect Prime Factorization: Ensure that you correctly factorize 48 into its prime components. The correct prime factorization of 48 is 2 × 2 × 2 × 2 × 3, or 2⁴ × 3. Missing or incorrect factors can lead to wrong simplifications.
• Forgetting to Pair Factors: When simplifying the square root, you need to pair the prime factors. For √48, pair the factors as follows: √(2⁴ × 3) = √(2⁴) × √3 = 4√3.
• Mistaking the Result: The simplified form of √48 is 4√3. Sometimes, people might mistakenly leave it as √48 or simplify it incorrectly. Ensure the final form is correct.
• Ignoring Decimal Form: The square root of 48 can also be expressed in decimal form, approximately 6.93. Both forms (4√3 and 6.93) represent the same value but in different notations.
• Confusing Radical Rules: Remember the product rule for radicals: √(a × b) = √a × √b. Apply this rule correctly to simplify √48.
• Overlooking Exponent Form: In exponent form, the square root of 48 is expressed as 48^1/2. This is another valid representation that should not be confused with incorrect notations.

By keeping these common mistakes in mind and understanding the clarifications, you can accurately simplify the square root of 48.