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Square Root of 689

The square root of 689 is expressed as √689 in the radical form and as (689)^½ or (689)^0.5 in the exponent form. The square root of 689 rounded up to 7 decimal places is 26.2488095. It is the positive solution of the equation x² = 689.

Square Root of 689: 26.248809496813376
Square Root of 689 in exponential form: (689)^½ or (689)^0.5
Square Root of 689 in radical form: √689

Square Root of 689

1.	What is the Square Root of 689?
2.	How to find the Square Root of 689?
3.	Is the Square Root of 689 Irrational?
4.	FAQs

What is the Square Root of 689?

The square root of 689, (or root 689), is the number which when multiplied by itself gives the product as 689. Therefore, the square root of 689 = √689 = 26.248809496813376.

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How to Find Square Root of 689?

Square Root of 689 by Long Division Method

Value of √689 by Long Division Method

Explanation:

Forming pairs: 06 and 89
Find a number Y (2) such that whose square is <= 6. Now divide 06 by 2 with quotient as 2.
Bring down the next pair 89, to the right of the remainder 2. The new dividend is now 289.
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 6) such that 4Z × Z <= 289. After finding Z, together 4 and Z (6) form a new divisor 46 for the new dividend 289.
Divide 289 by 46 with the quotient as 6, giving the remainder = 289 - 46 × 6 = 289 - 276 = 13.
Now, let's find the decimal places after the quotient 26.
Bring down 00 to the right of this remainder 13. The new dividend is now 1300.
Add the last digit of quotient to divisor i.e. 6 + 46 = 52. To the right of 52, find a digit Z (which is 2) such that 52Z × Z <= 1300. Together they form a new divisor (522) for the new dividend (1300).
Divide 1300 by 522 with the quotient as 2, giving the remainder = 1300 - 522 × 2 = 1300 - 1044 = 256.
Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 689.

Therefore, the square root of 689 by long division method is 26.2 approximately.

Is Square Root of 689 Irrational?

The actual value of √689 is undetermined. The value of √689 up to 25 decimal places is 26.24880949681337452637319. Hence, the square root of 689 is an irrational number.

☛ Also Check:

Square Root of 30 - √30 = 5.47723
Square Root of 1024 - √1024 = 32
Square Root of 109 - √109 = 10.44031
Square Root of 8 - √8 = 2.82843
Square Root of 65 - √65 = 8.06226
Square Root of 42 - √42 = 6.48074
Square Root of 27 - √27 = 5.19615

Square Root of 689 Solved Examples

Example 1: Solve the equation x² − 689 = 0

Solution:

x² - 689 = 0 i.e. x² = 689
x = ±√689
Since the value of the square root of 689 is 26.249,
⇒ x = +√689 or -√689 = 26.249 or -26.249.
Example 2: If the surface area of a cube is 4134 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 = 4134 in²
⇒ a = ±√689 in
Since length can't be negative,
⇒ a = √689
We know that the square root of 689 is 26.249.
⇒ a = 26.249 in
Example 3: If the surface area of a sphere is 2756π in². Find the radius of the sphere.

Solution:

Let 'r' be the radius of the sphere.
⇒ Area of the sphere = 4πr² = 2756π in²
⇒ r = ±√689 in
Since radius can't be negative,
⇒ r = √689
The square root of 689 is 26.249.
⇒ r = 26.249 in

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FAQs on the Square Root of 689

What is the Value of the Square Root of 689?

The square root of 689 is 26.2488.

Why is the Square Root of 689 an Irrational Number?

Upon prime factorizing 689 i.e. 13¹ × 53¹, 13 is in odd power. Therefore, the square root of 689 is irrational.

What is the Square Root of -689?

The square root of -689 is an imaginary number. It can be written as √-689 = √-1 × √689 = i √689 = 26.248i
where i = √-1 and it is called the imaginary unit.

If the Square Root of 689 is 26.249. Find the Value of the Square Root of 6.89.

Let us represent √6.89 in p/q form i.e. √(689/100) = 6.89/10 = 2.625. Hence, the value of √6.89 = 2.625

Is the number 689 a Perfect Square?

The prime factorization of 689 = 13¹ × 53¹. Here, the prime factor 13 is not in the pair. Therefore, 689 is not a perfect square.

What is the Square of the Square Root of 689?

The square of the square root of 689 is the number 689 itself i.e. (√689)² = (689)^2/2 = 689.

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