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Square Root of 312

The square root of 312 is expressed as √312 in the radical form and as (312)^½ or (312)^0.5 in the exponent form. The square root of 312 rounded up to 8 decimal places is 17.66352173. It is the positive solution of the equation x² = 312. We can express the square root of 312 in its lowest radical form as 2 √78.

Square Root of 312: 17.663521732655695
Square Root of 312 in exponential form: (312)^½ or (312)^0.5
Square Root of 312 in radical form: √312 or 2 √78

Square Root of 312

1.	What is the Square Root of 312?
2.	How to find the Square Root of 312?
3.	Is the Square Root of 312 Irrational?
4.	FAQs

What is the Square Root of 312?

The square root of 312, (or root 312), is the number which when multiplied by itself gives the product as 312. Therefore, the square root of 312 = √312 = 2 √78 = 17.663521732655695.

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How to Find Square Root of 312?

Square Root of 312 by Long Division Method

Value of √312 by Long Division Method

Explanation:

Forming pairs: 03 and 12
Find a number Y (1) such that whose square is <= 3. Now divide 03 by 1 with quotient as 1.
Bring down the next pair 12, to the right of the remainder 2. The new dividend is now 212.
Add the last digit of the quotient (1) to the divisor (1) i.e. 1 + 1 = 2. To the right of 2, find a digit Z (which is 7) such that 2Z × Z <= 212. After finding Z, together 2 and Z (7) form a new divisor 27 for the new dividend 212.
Divide 212 by 27 with the quotient as 7, giving the remainder = 212 - 27 × 7 = 212 - 189 = 23.
Now, let's find the decimal places after the quotient 17.
Bring down 00 to the right of this remainder 23. The new dividend is now 2300.
Add the last digit of quotient to divisor i.e. 7 + 27 = 34. To the right of 34, find a digit Z (which is 6) such that 34Z × Z <= 2300. Together they form a new divisor (346) for the new dividend (2300).
Divide 2300 by 346 with the quotient as 6, giving the remainder = 2300 - 346 × 6 = 2300 - 2076 = 224.
Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 312.

Therefore, the square root of 312 by long division method is 17.6 approximately.

Is Square Root of 312 Irrational?

The actual value of √312 is undetermined. The value of √312 up to 25 decimal places is 17.66352173265569370952809. Hence, the square root of 312 is an irrational number.

☛ Also Check:

Square Root of 108 - √108 = 10.39230
Square Root of 729 - √729 = 27
Square Root of 92 - √92 = 9.59166
Square Root of 65 - √65 = 8.06226
Square Root of 41 - √41 = 6.40312
Square Root of 320 - √320 = 17.88854
Square Root of 30 - √30 = 5.47723

Square Root of 312 Solved Examples

Example 1: Solve the equation x² − 312 = 0

Solution:

x² - 312 = 0 i.e. x² = 312
x = ±√312
Since the value of the square root of 312 is 17.664,
⇒ x = +√312 or -√312 = 17.664 or -17.664.
Example 2: If the surface area of a cube is 1872 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 = 1872 in²
⇒ a = ±√312 in
Since length can't be negative,
⇒ a = √312
We know that the square root of 312 is 17.664.
⇒ a = 17.664 in
Example 3: If the area of an equilateral triangle is 312√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 = 312√3 in²
⇒ a = ±√1248 in
Since length can't be negative,
⇒ a = √1248 = 2 √312
We know that the square root of 312 is 17.664.
⇒ a = 35.327 in

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FAQs on the Square Root of 312

What is the Value of the Square Root of 312?

The square root of 312 is 17.66352.

Why is the Square Root of 312 an Irrational Number?

Upon prime factorizing 312 i.e. 2³ × 3¹ × 13¹, 2 is in odd power. Therefore, the square root of 312 is irrational.

What is the Square Root of -312?

The square root of -312 is an imaginary number. It can be written as √-312 = √-1 × √312 = i √312 = 17.663i
where i = √-1 and it is called the imaginary unit.

What is the Value of 16 square root 312?

The square root of 312 is 17.664. Therefore, 16 √312 = 16 × 17.664 = 282.616.

What is the Square of the Square Root of 312?

The square of the square root of 312 is the number 312 itself i.e. (√312)² = (312)^2/2 = 312.

What is the Square Root of 312 in Simplest Radical Form?

We need to express 312 as the product of its prime factors i.e. 312 = 2 × 2 × 2 × 3 × 13. Therefore, √312 = √2 × 2 × 2 × 3 × 13 = 2 √78. Thus, the square root of 312 in the lowest radical form is 2 √78.

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