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 âˆš48.
Hence, âˆš48 = âˆš(Number Ã— Number)
Thus, if we multiply the number 6.928 two times, we get the original value 48.
(i.e) âˆš48 = âˆš(6.928Ã— 6.928)
âˆš48 = âˆš(6.928)^{2}
Now, remove square and square root, we get
âˆš48 = Â± 6.928 (Rounded to three decimal places)
Square Root of 48 in Decimal Form: 6.928. |
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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 Ã— 2 Ã— 2 Ã— 2 Ã— 3, which is equal to 2^{4} Ã— 3. Thus, the square root of 48 in radical form is expressed as follows:
âˆš48 = âˆš(2 Ã— 2 Ã— 2 Ã— 2 Ã— 3)
âˆš48 = âˆš(2)^{2}.âˆš(2)^{2} âˆš3.
âˆš 48 = 2 Ã— 2 Ã—âˆš3
âˆš48 = 4âˆš3.
Therefore, the square root of 48 in radical form is 4âˆš3.
Square Root of 48 in Radical Form: 4âˆš3. |
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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 Ã— 2 Ã— 2 Ã— 2 Ã— 3.
Thus, âˆš48 = âˆš(2 Ã— 2 Ã— 2 Ã— 2 Ã— 3) = 4âˆš3
We know that, the value of âˆš3 is 1.732.
Now, substitute the value in the above expression, we get
âˆš48 = 4(1.732)
âˆš48 = 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, 6Ã—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) Ã— new number should give the product value, that should be less than or equal to 1200. Hence, (120+9) Ã— 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) Ã— new number should give the product value, that should be less than or equal to 3900. Thus, (1380+2)Ã— 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, âˆš48 is 6.928.
Learn More on Square Root of a Number: |
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Video Lessons on Square Roots
Visualising square roots
Finding Square roots
Examples
Example 1:
Simplify (âˆš3âˆš48) – 2
Solution:
Given: (âˆš3âˆš48) – 2â€¦(1)
We know that the square root of 48 in radical form is 4âˆš3.
Now, substitute âˆš48 = 4âˆš3 in (1) we get,
(âˆš3âˆš48) – 2= (âˆš3. 4âˆš3) – 2
(âˆš3âˆš48) – 2= 12- 2
(âˆš3âˆš48) – 2= 10
Therefore, (âˆš3âˆš48) – 2 = 10
Example 2:
Find the value of m, if 4m + âˆš48 = 12
Solution:
Given equation: 4m + âˆš48 = 12â€¦(1)
We know that âˆš48 = 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.
Frequently Asked Questions on Square Root of 48
What is the value of the square root of 48?
The value of the square root of 48 is approximately equal to 6.928, which is rounded to three decimal places.
What is the square root of 48 in radical form?
The square root of 48 in radical form is 4âˆš3.
Is the square root of 48 a rational number?
No, the square root of 48 is not a rational number, as it cannot be expressed in the p/q form.
Is 48 a perfect square number?
No, 48 is not a perfect square number, since it cannot be written as the product of two equal numbers.
Is 48 a perfect cube?
No, 48 is not a perfect cube, since it cannot be written as the product of three equal integers.