Square Root of 392

The square root of 392 can be determined using different methods. A few methods that give a close approximation of the square root of 392 will be discussed in this article. The square root of 392 is an irrational number because 392 is not a perfect square number. A perfect square number is a number that can be written as a square of an integer. The square root of a number is just the opposite of squaring. In radical form, the square root of 392 is denoted as √392, and in the exponential form, it is written as (392)^½. Let us find the square root of 392.

Learn more about square and square roots.

What Is the Square Root of 392?

The square root of 392 is a number whose square gives the result 392. Now, 392 is not a perfect square number because we cannot find any integer that could be multiplied twice to get 392. Thus, we find an approximate value of the square root of 392 as it is an irrational number.

The square root of 392 can also be determined by finding the roots of the quadratic equation

x² – 392 = 0

⇒ x² = 392

Taking square roots on both sides, we get,

⇒ x = √392

⇒ x = ± 14√2 ≈ 19.798 (approx.)

Learn more about the properties of perfect squares.

How to Find the Square Root of 392?

Let us find the square root of 392 using the prime factorisation method, Newton’s formula, and the long division method. The repeated subtraction method will not work as 392 is not a perfect square, and its square root is an irrational number.

Prime Factorisation Method

To find the square root of 392 by the prime factorisation method, we first prime factorise 392 and then make pairs of two to get the square root.

Prime factorisation of 392 = 2 × 2 × 2 × 7 × 7

For each pair of prime factors of 392, we take that factor once to determine the square root of 392.

∴ the square root of 392 = √392 = √(2 × 2 × 2 × 7 × 7) = 2 × 7 × √2 = 14 √2

As √2 is an irrational number and cannot be simplified further using the prime factorisation method, the square root of 392 is an irrational number.

Newton’s Formula

Newton’s formula to determine the approximate square root of any number (also referred to as the approximation method of finding the square root) is given by

√N ≈ ½ (N/A + A), where

N is the number whose square root we need to find.

A is the approximate guess square root number. The guess square root could be the greatest number whose square is less than or equal to 392. We can also take the smallest number whose square is greater than or equal to 392

Now, N = 392 and 19² = 361 < 392, we take A = 19

Let us substitute these values in the formula

√392 ≈ ½ (392/19 + 19) ≈ 19.8, which is nearly equal to the actual square root of 392.

Long Division Method

To find the square root of 392 by the long division method, we shall write 392 as the dividend and pair its digits from right to left. We shall calculate the square root as follows:

• While pairing from the right side, 3 remains unpaired. We put a pair of zeros after the number. Remember, whenever we go to the next step of division, both the paired numbers will come down together.
• Starting with 3 as the first dividend, now 1² < 3 < 2². Therefore, the square of 1 goes into 3, taking 1 as the quotient.
• For the next step, 1 + 1 = 2 becomes the first part of the divisor, and the remainder of 3 – 1 = 2, along with 92, that is 292, becomes the next dividend.
• Now, 29 × 9 = 261 < 292, thus we divide 292 by 29, taking 9 as a quotient.
• For the next step, the first part of the divisor will be 29 + 9 = 38, and the dividend will be the remainder from 292 – 261 = 31 along with a pair of zeros, that is, 3100.
• Proceed in the same way we get the square root of 392.

Note: The last divisor is always double the quotient. Using this fact, we can always verify our answer.

Thus, the square root of 392 by the long division method is 19.798.

To learn more about 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 392

Example 1:

Find the semi-perimeter of the circle whose area is 392𝜋 m².

Solution:

Let r be the radius of the circle.

The area of the circle = 𝜋r² = 392𝜋 m²

⇒ r² = 392 (taking square root on both sides)

⇒ r = √392

⇒ r = 14√2 m (taking the positive root as length cannot be negative)

∴ the semi-perimeter of the circle = 𝜋r = 22/7 × 14√2 = 44√2 m

Example 2:

Find the length of the base of a triangle whose area is 39.2 cm² and the height is one-fifth of its base.

Solution:

Let b be the base of the triangle.

Height of the triangle = b/5

Area of the triangle = ½ × b × b/5 = 39.2 cm²

⇒ 1/10 × b² = 39.2

⇒ b² = 39.2 × 10 = 392 (taking square root on both sides)

⇒ b = √392 ≈ 19.79 cm

∴ the length of the base of the triangle is 19.79 cm.

Example 3:

What is the smallest number that should be subtracted from 392 to make it a perfect square number? Also, find the square root of that number obtained.

Solution:

Clearly, 19² = 361 < 392 is the greatest perfect square number less than 392.

Therefore, we must subtract 31 from 392 to make it a perfect square number.

Then, 392 – 31 = 361 is a perfect square number and

√361 = ± 19