Square Root of 48

In Mathematics, a number, which when multiplied by itself results in the original number 48 is called the square root of 48. The square root of 48 is considered to be an irrational number since it cannot be expressed in the form of p/q and the decimal part of the value of the square root of 48 is never-ending. There are two different ways to find the square root of 48, such as prime factorization and the long division method. Here, we will discuss how to find the square root 48 using these two methods with complete explanation.

Table of Contents:

• What Is The Square Root Of 48?
• Square Root Of 48 In Radical Form
• Square Root Of 48 By Prime Factorization Method
• Square Root Of 48 By Long Division Method
• Solved Examples
• FAQs

What is the Square Root of 48?

The square root of 48 is a number, which when multiplied by itself and gives the result as 48. The square root of 48 is symbolically expressed as √48.

Hence, √48 = √(Number × Number)

Thus, if we multiply the number 6.928 two times, we get the original value 48.

(i.e) √48 = √(6.928× 6.928)

√48 = √(6.928)²

Now, remove square and square root, we get

√48 = ± 6.928 (Rounded to three decimal places)

Square Root of 48 in Radical Form

To find the radical form of the square root of 48, we should know the prime factorization of 48. Thus, the prime factorization of 48 is 2 × 2 × 2 × 2 × 3, which is equal to 2⁴ × 3. Thus, the square root of 48 in radical form is expressed as follows:

√48 = √(2 × 2 × 2 × 2 × 3)

√48 = √(2)².√(2)² √3.

√ 48 = 2 × 2 ×√3

√48 = 4√3.

Therefore, the square root of 48 in radical form is 4√3.

Square Root of 48 by Prime Factorization Method

The square root of 48 is found using two different methods, such as the prime factorization method and the long division method. Now, let us discuss the prime factorization method to find the square root of 48. To find the square root of 48 using the prime factorization method, one must know the prime factorization of 48. We know that the prime factorization of 48 is 2 × 2 × 2 × 2 × 3.

Thus, √48 = √(2 × 2 × 2 × 2 × 3) = 4√3

We know that, the value of √3 is 1.732.

Now, substitute the value in the above expression, we get

√48 = 4(1.732)

√48 = 6.928 (approximately)

Hence, the square root of 48 in decimal form is approximately equal to 6.928 (rounded to three decimal places)

Square Root of 48 by Long Division Method

Follow the below steps to find the square root of 48 using the long division method:

Step 1: Write the number 48 in decimal form. To find the exact value of the square root of 48, add 6 zeros after the decimal point. Hence, 48 in decimal form is 48.000000. Now, pair the number from right to left by putting the bar on the top of the number.

Step 2: Now, divide the number 48 by a number, such that the product of the same number should be less than or equal to 48. Thus, 6×6 =36, which is less than 48. Thus, we obtained the quotient = 6 and remainder = 12.

Step 3: Double the quotient value, so we get 12, and assume that 120 is the new divisor. Now, bring down the value 00 for division operation. So, the new dividend obtained is 1200. Now, find the number, such that (120 + new number) × new number should give the product value, that should be less than or equal to 1200. Hence, (120+9) × 9 = 1161, which is less than 1200.

Step 4: Now subtract 1161 from 1200, and we get 39 as the new reminder, and 69 as a quotient.

Step 5: The new quotient obtained is 69, and double that. Hence, we get 138 and assume that 1380 is our new divisor. Now, bring down the two zeros and, we have 3900 as the new dividend.

Step 6: Find the number, such that (1380 + new number) × new number should give the product value, that should be less than or equal to 3900. Thus, (1380+2)× 2 = 2764, which is less than 3900.

Step 7: Subtract 2764 from 3900, and we get 1136 as the new reminder.

Step 8: Continue this process until we get the approximate value of the square root of 48 up to three decimal places. (Note: keep the decimal point in the quotient value after bringing down all the values in the dividend).

Step 9: Thus, the approximate value of the square root of 48, √48 is 6.928.

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Examples

Example 1:

Simplify (√3√48) – 2

Solution:

Given: (√3√48) – 2…(1)

We know that the square root of 48 in radical form is 4√3.

Now, substitute √48 = 4√3 in (1) we get,

(√3√48) – 2= (√3. 4√3) – 2

(√3√48) – 2= 12- 2

(√3√48) – 2= 10

Therefore, (√3√48) – 2 = 10

Example 2:

Find the value of m, if 4m + √48 = 12

Solution:

Given equation: 4m + √48 = 12…(1)

We know that √48 = 6.928.

Now, substitute the value in equation (1),

4m + 6.928 = 12

4m = 12 – 6.928

4m = 5.072

m = 5.072/4

m = 1.268

Therefore, the value of m is 1.268.