The square of root 441 is 21. The square root of a number is a number that, when multiplied two times to itself, gives the perfect square number. The square root of 441 is written as √441, where ‘√’ is called the radical sign. The square root of 441 in exponential form is written as (441)½ . Since the square root of 441 is a whole number, therefore 441 is a perfect square number. In this article, we shall learn how to find the square root of 441.
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Square Root of 441 |
± 21 |
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Square of 441 |
194,481 |
.What is the Square Root of 441?
The square root of 441 is a number whose square is 441. Now, if we square both 21 and –21, we get the answer 441. Thus, the square root of 441 is both 21 and –21. We choose the positive or the negative root as per the requirement of the problem.
We can also express the square root of 441 as the roots of the quadratic equation x2 – 441 = 0
x2 – 441 = 0
⇒ x2 = 441 (taking square roots on both sides)
⇒ x = √441
⇒ x = ± 21
Check out the properties of perfect square numbers.
How to Find the Square Root of 441?
Let us calculate the square root of 441 using different methods:
- Repeated subtraction method
- Prime factorisation method
- Long division method
Repeated Subtraction Method
Square of any perfect square number can be determined by the repeated subtraction method. In this method, we have to repeatedly subtract the given number by consecutive odd numbers until we get the answer zero. The nth odd number for which we get the answer zero is the square root of the given number.
Thus, we have 441 – 1 = 440, 440 – 3 = 437, 437 – 5 = 432, continuing in this way we get for the 21st odd number the answer is zero.
∴ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 + 35 + 37 + 39 + 41 = 441.
This method is quite a time-consuming method, and for larger numbers, this method is not preferable. Instead, we can use the prime factorisation method to calculate the square roots.
Prime Factorisation Method
We shall prime factorise the given, then make pairs of two for each number to find the square root of the number.
Prime factorisation of 441 = 3 × 3 × 7 × 7
√441 = √(3 × 3 × 7 × 7) = 3 × 7 = 21.
Thus, to find the square root of any number by the prime factorisation method, the following are steps:
- Prime factorise the given number
- Make pairs of two for each of the prime factors.
- Take only one prime factor for each pair.
- If any prime remains unpaired, then the number is not a perfect square.
Long Division Method
To calculate the square root of 441 by the long division method, we make pairs of digits of 441 from right to left. Then perform the division as follows:
To learn how to find the square root of any number by the long division method, click here.
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Solved Examples on Square Root of 441
Example 1:
Find the smallest number which must be multiplied to 147 to make it a perfect square number. Also, find the square root of the perfect square number.
Solution:
Prime factorisation of 147 = 3 × 7 × 7
We observe that only 3 remains unpaired, thus, 3 must be multiplied to 147.
147 × 3 = 441 is the perfect square number.
Square root of 441 = √[3 × 3 × 7 × 7] = 3 × 7 = 21.
Example 2:
Find the length of the third side of a right-angled triangle whose length of the longest side is 29 cm and the length of one of the perpendicular sides is 20 cm.
Solution:
Let the length of the third side be x.
Then, 292 = x2 + 202
⇒ x2 = 841 – 400 = 441
⇒ x = √441
⇒ x = 21 cm (taking positive square root)
Therefore, length of the third side is 21 cm.
Example 3:
Find the diameter of the sphere whose total surface area is 5538.96 cm2. (Use 𝜋 = 3.14)
Solution:
Let r be the radius of the sphere.
The total surface area of the sphere = 4𝜋r2 = 5538.96 cm2
⇒ r2 = (5538.96)/( 4 × 3.14) = 441
⇒ r = √441 = 21 cm
∴ Diameter of the sphere is 2 × 21 = 42 m.
Frequently Asked Questions on Square Root of 441
What is the square root 441?
The square root of 441 is ± 21.
Is 441 a perfect square number?
Yes, 441 is a perfect square number and 21 × 21 = 441.
What is the prime factorisation of 441?
The prime factorisation of 441 is 3 × 3 × 7 × 7.
How many consecutive odd numbers must be added to get 441?
The sum of the first 21 consecutive odd numbers is 441. Thus, √441 = 21.
How do you find the square root of 441?
We can find the square root of 441 by the prime factorisation method. The prime factorisation of 441 = 3 × 3 × 7 × 7. By making the pairs of two, the square root of 441 = √[3 × 3 × 7 × 7] = 3 × 7 = 21.
What is the cube root of 441?
The cube root of 441 is 7.611662 (approx.).
