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Square Root of 409

The square root of 409 is expressed as √409 in the radical form and as (409)^½ or (409)^0.5 in the exponent form. The square root of 409 rounded up to 5 decimal places is 20.22375. It is the positive solution of the equation x² = 409.

Square Root of 409: 20.223748416156685
Square Root of 409 in exponential form: (409)^½ or (409)^0.5
Square Root of 409 in radical form: √409

Square Root of 409

1.	What is the Square Root of 409?
2.	How to find the Square Root of 409?
3.	Is the Square Root of 409 Irrational?
4.	FAQs

What is the Square Root of 409?

The square root of 409, (or root 409), is the number which when multiplied by itself gives the product as 409. Therefore, the square root of 409 = √409 = 20.223748416156685.

☛ Check: Square Root Calculator

How to Find Square Root of 409?

Square Root of 409 by Long Division Method

Value of √409 by Long Division Method

Explanation:

Forming pairs: 04 and 09
Find a number Y (2) such that whose square is <= 4. Now divide 04 by 2 with quotient as 2.
Bring down the next pair 09, to the right of the remainder 0. The new dividend is now 9.
Add the last digit of the quotient (2) to the divisor (2) i.e. 2 + 2 = 4. To the right of 4, find a digit Z (which is 0) such that 4Z × Z <= 9. After finding Z, together 4 and Z (0) form a new divisor 40 for the new dividend 9.
Divide 9 by 40 with the quotient as 0, giving the remainder = 9 - 40 × 0 = 9 - 0 = 9.
Now, let's find the decimal places after the quotient 20.
Bring down 00 to the right of this remainder 9. The new dividend is now 900.
Add the last digit of quotient to divisor i.e. 0 + 40 = 40. To the right of 40, find a digit Z (which is 2) such that 40Z × Z <= 900. Together they form a new divisor (402) for the new dividend (900).
Divide 900 by 402 with the quotient as 2, giving the remainder = 900 - 402 × 2 = 900 - 804 = 96.
Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 409.

Therefore, the square root of 409 by long division method is 20.2 approximately.

Is Square Root of 409 Irrational?

The actual value of √409 is undetermined. The value of √409 up to 25 decimal places is 20.22374841615668437951299. Hence, the square root of 409 is an irrational number.

☛ Also Check:

Square Root of 63 - √63 = 7.93725
Square Root of 325 - √325 = 18.02776
Square Root of 841 - √841 = 29
Square Root of 240 - √240 = 15.49193
Square Root of 8 - √8 = 2.82843
Square Root of 44 - √44 = 6.63325
Square Root of 106 - √106 = 10.29563

Square Root of 409 Solved Examples

Example 1: Solve the equation x² − 409 = 0

Solution:

x² - 409 = 0 i.e. x² = 409
x = ±√409
Since the value of the square root of 409 is 20.224,
⇒ x = +√409 or -√409 = 20.224 or -20.224.
Example 2: If the area of a circle is 409π in². Find the radius of the circle.

Solution:

Let 'r' be the radius of the circle.
⇒ Area of the circle = πr² = 409π in²
⇒ r = ±√409 in
Since radius can't be negative,
⇒ r = √409
The square root of 409 is 20.224.
⇒ r = 20.224 in
Example 3: If the surface area of a cube is 2454 in². Find the length of the side of the cube.

Solution:

Let 'a' be the length of the side of the cube.
⇒ Area of the cube = 6a² = 2454 in²
⇒ a = ±√409 in
Since length can't be negative,
⇒ a = √409
We know that the square root of 409 is 20.224.
⇒ a = 20.224 in

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FAQs on the Square Root of 409

What is the Value of the Square Root of 409?

The square root of 409 is 20.22374.

Why is the Square Root of 409 an Irrational Number?

The number 409 is prime. This implies that the number 409 is pairless and is not in the power of 2. Therefore, the square root of 409 is irrational.

What is the Square Root of -409?

The square root of -409 is an imaginary number. It can be written as √-409 = √-1 × √409 = i √409 = 20.223i
where i = √-1 and it is called the imaginary unit.

Evaluate 16 plus 4 square root 409

The given expression is 16 + 4 √409. We know that the square root of 409 is 20.224. Therefore, 16 + 4 √409 = 16 + 4 × 20.224 = 16 + 80.895 = 96.895

Is the number 409 a Perfect Square?

The number 409 is prime. This implies that the square root of 409 cannot be expressed as a product of two equal integers. Therefore, the number 409 is not a perfect square.

What is the Value of 15 square root 409?

The square root of 409 is 20.224. Therefore, 15 √409 = 15 × 20.224 = 303.356.

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