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Square Root of 772

The square root of 772 is expressed as √772 in the radical form and as (772)^½ or (772)^0.5 in the exponent form. The square root of 772 rounded up to 8 decimal places is 27.78488798. It is the positive solution of the equation x² = 772. We can express the square root of 772 in its lowest radical form as 2 √193.

Square Root of 772: 27.784887978899608
Square Root of 772 in exponential form: (772)^½ or (772)^0.5
Square Root of 772 in radical form: √772 or 2 √193

Square Root of 772

1.	What is the Square Root of 772?
2.	How to find the Square Root of 772?
3.	Is the Square Root of 772 Irrational?
4.	FAQs

What is the Square Root of 772?

The square root of 772, (or root 772), is the number which when multiplied by itself gives the product as 772. Therefore, the square root of 772 = √772 = 2 √193 = 27.784887978899608.

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How to Find Square Root of 772?

Square Root of 772 by Long Division Method

Value of √772 by Long Division Method

Explanation:

Forming pairs: 07 and 72
Find a number Y (2) such that whose square is <= 7. Now divide 07 by 2 with quotient as 2.
Bring down the next pair 72, to the right of the remainder 3. The new dividend is now 372.
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 7) such that 4Z × Z <= 372. After finding Z, together 4 and Z (7) form a new divisor 47 for the new dividend 372.
Divide 372 by 47 with the quotient as 7, giving the remainder = 372 - 47 × 7 = 372 - 329 = 43.
Now, let's find the decimal places after the quotient 27.
Bring down 00 to the right of this remainder 43. The new dividend is now 4300.
Add the last digit of quotient to divisor i.e. 7 + 47 = 54. To the right of 54, find a digit Z (which is 7) such that 54Z × Z <= 4300. Together they form a new divisor (547) for the new dividend (4300).
Divide 4300 by 547 with the quotient as 7, giving the remainder = 4300 - 547 × 7 = 4300 - 3829 = 471.
Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 772.

Therefore, the square root of 772 by long division method is 27.7 approx.

Is Square Root of 772 Irrational?

The actual value of √772 is undetermined. The value of √772 up to 25 decimal places is 27.78488797889960901686509. Hence, the square root of 772 is an irrational number.

☛ Also Check:

Square Root of 512 - √512 = 22.62742
Square Root of 22 - √22 = 4.69042
Square Root of 49 - √49 = 7
Square Root of 200 - √200 = 14.14214
Square Root of 54 - √54 = 7.34847
Square Root of 149 - √149 = 12.20656
Square Root of 120 - √120 = 10.95445

Square Root of 772 Solved Examples

Example 1: Solve the equation x² − 772 = 0

Solution:

x² - 772 = 0 i.e. x² = 772
x = ±√772
Since the value of the square root of 772 is 27.785,
⇒ x = +√772 or -√772 = 27.785 or -27.785.
Example 2: If the area of an equilateral triangle is 772√3 in². Find the length of one of the sides of the triangle.

Solution:

Let 'a' be the length of one of the sides of the equilateral triangle.
⇒ Area of the equilateral triangle = (√3/4)a² = 772√3 in²
⇒ a = ±√3088 in
Since length can't be negative,
⇒ a = √3088 = 2 √772
We know that the square root of 772 is 27.785.
⇒ a = 55.570 in
Example 3: If the surface area of a sphere is 3088π in². Find the radius of the sphere.

Solution:

Let 'r' be the radius of the sphere.
⇒ Area of the sphere = 4πr² = 3088π in²
⇒ r = ±√772 in
Since radius can't be negative,
⇒ r = √772
The square root of 772 is 27.785.
⇒ r = 27.785 in

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FAQs on the Square Root of 772

What is the Value of the Square Root of 772?

The square root of 772 is 27.78488.

Why is the Square Root of 772 an Irrational Number?

Upon prime factorizing 772 i.e. 2² × 193¹, 193 is in odd power. Therefore, the square root of 772 is irrational.

Is the number 772 a Perfect Square?

The prime factorization of 772 = 2² × 193¹. Here, the prime factor 193 is not in the pair. Therefore, 772 is not a perfect square.

Evaluate 3 plus 11 square root 772

The given expression is 3 + 11 √772. We know that the square root of 772 is 27.785. Therefore, 3 + 11 √772 = 3 + 11 × 27.785 = 3 + 305.634 = 308.634

What is the Square Root of 772 in Simplest Radical Form?

We need to express 772 as the product of its prime factors i.e. 772 = 2 × 2 × 193. Therefore, √772 = √2 × 2 × 193 = 2 √193. Thus, the square root of 772 in the lowest radical form is 2 √193.

What is the Square Root of -772?

The square root of -772 is an imaginary number. It can be written as √-772 = √-1 × √772 = i √772 = 27.784i
where i = √-1 and it is called the imaginary unit.

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