Square Roots (वर्गमूल) In Hindi
किसी भी दी गई संख्या का वर्गमूल वह संख्या होती है, जिसका वर्ग दी गई संख्या ही होती है। एक वर्गमूल को (√) चिह्न द्वारा दर्शाया जाता है। यदि \( x^2 \) एक वर्ग संख्या है तो x उसका वर्गमूल है।
उदाहरण:
1. 100 का वर्गमूल 10 है क्योंकि 10² = 100
2. 16 का वर्गमूल 4 है क्योंकि 4² = 16
3. 9 का वर्गमूल 3 है क्योंकि 3² = 9
Square Roots In English
Square root of a number is a value, which on multiplication by itself, gives the original number. The square root is an inverse method of squaring a number. Hence, squares and square roots are related concepts.
In other words, if we say “the square root of x is y,” it means that:
\( y^2 \) = x
Here, “y” is the square root, and “x” is the number we’re taking the square root of.
Here are some examples:
1. The square root of 9 is 3, because 3² = 9.
2. The square root of 16 is 4, because 4² = 16.
3. The square root of 25 is 5, because 5² = 25.
The methods to find the square root of numbers are:
1. Square Root by Prime Factorizations
2. Square Root by Repeated Subtraction Method
3. Square Root by Long Division Method
4. Square Root by Estimation Method
Square Root by Prime Factorizations-
The square root by prime factorization method defines and finds the square root of a number by analyzing its prime factors and their pairing pattern. It’s specifically applicable to perfect squares, which are integers resulting from squaring another integer. Therefore, the square root of 36 is 6, confirming 36 is a perfect square (6 x 6 = 36).
Square Root by Repeated Subtraction Method-
The square root by repeated subtraction method is an older technique for finding the square root of a number, specifically perfect squares. It relies on the fact that the sum of the first n consecutive odd numbers equals n squared n².
Example:
Find the square root of 25 using the repeated subtraction method.
Start with 25.
Subtract 1: 25 – 1 = 24
Subtract 3: 24 – 3 = 21
Subtract 5: 21 – 5 = 16
Subtract 7: 16 – 7 = 9
Subtract 9: 9 – 9 = 0
We subtracted five consecutive odd numbers (1, 3, 5, 7, and 9) before reaching zero.
Therefore, the square root of 25 is 5.
Important notes:
This method is only efficient for small perfect squares.
Square Root by Long Division Method-
The long division method for square roots is a systematic algorithm used to find the square root of any number, not just perfect squares. It resembles the familiar long division process for regular division, but with some key adaptations.
Example:
Find the square root of 128 using the long division method.
4
——-
128 | 16 (4² = 16, so 4 is the first digit of the quotient)
– 16
—–
08
8
——-
00
– 00
—–
0
Therefore, the square root of 128 is 4.√128.
Square Root by Estimation Method-
The square root by estimation method is a technique to approximate the square root of a number, particularly useful for non-perfect squares or quick estimations. It relies on identifying nearby perfect squares and comparing the given number to them.
Example:
Estimate the square root of 32 using the estimation method.
Nearest perfect squares: 25 (5²) and 36 (6²)
Initial estimate: √25 = 5
Since 32 is closer to 36 than 25, our initial estimate (5) is slightly low.
Refined estimate: As 32 is closer to 36, the actual square root will be slightly higher than 5. A common adjustment for such situations is adding half the difference between the two square roots: (6 – 5) / 2 = 0.5.
Therefore, the estimated square root of 32 is approximately 5.5.
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