Square Root of 390
2026-02-28 01:06 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse operation of squaring is finding the square root. The square root is used in fields like vehicle design, finance, etc. Here, we will discuss the square root of 390.

What is the Square Root of 390?

The square root is the inverse of squaring a number. 390 is not a perfect square. The square root of 390 can be expressed in both radical and exponential forms. In radical form, it is expressed as √390, whereas in exponential form, it is (390)^(1/2). √390 ≈ 19.748, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 390

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 390, the long division method and approximation method are more appropriate. Let us explore these methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 390 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Let's see how 390 is broken down into its prime factors.

Step 1: Finding the prime factors of 390 Breaking it down, we get 2 x 3 x 5 x 13: 2^1 x 3^1 x 5^1 x 13^1

Step 2: We found the prime factors of 390. The next step is to make pairs of those prime factors. Since 390 is not a perfect square, the digits of the number cannot be grouped into pairs, making it impossible to calculate the square root using prime factorization.

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Square Root of 390 by Long Division Method

The long division method is particularly useful for non-perfect square numbers. Let's learn how to find the square root using the long division method, step by step.

Step 1: To begin, group the numbers from right to left. In the case of 390, we group it as 90 and 3.

Step 2: Find n whose square is less than or equal to 3. We choose n = 1 because 1 x 1 ≤ 3. The quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 90 to get a new dividend of 290. Add the old divisor (1) to itself to get 2, the new divisor.

Step 4: Find n such that 2n x n ≤ 290. If n = 9, 29 x 9 = 261.

Step 5: Subtract 261 from 290 to get a remainder of 29

Step 6: Add a decimal point and bring down two zeros to make the new dividend 2900.

Step 7: Find the new divisor. Adding a digit to 58 yields 589. Choose n = 4, resulting in 589 x 4 = 2356.

Step 8: Subtract 2356 from 2900 to get 544.

Step 9: Continue the process until you reach the desired level of precision. The quotient is approximately 19.74.

So, the square root of √390 is approximately 19.74.

Square Root of 390 by Approximation Method

The approximation method is another useful approach to finding square roots. It is an easy way to estimate the square root of a number. Let's learn how to find the square root of 390 using the approximation method.

Step 1: Identify the closest perfect squares around 390.

The smallest perfect square less than 390 is 361, and the largest perfect square greater than 390 is 400.

√390 falls between 19 and 20.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

(390 - 361) / (400 - 361) ≈ 0.7436

Adding this decimal to the lower whole number, 19 + 0.7436 ≈ 19.74.

So, the approximate square root of 390 is 19.74.

Common Mistakes and How to Avoid Them in the Square Root of 390

Students often make mistakes when finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's explore some common mistakes in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √390?

Okay, lets begin

The area of the square is approximately 390 square units.

Explanation

The area of a square = side^2.

The side length is given as √390. Area = side^2 ≈ √390 x √390 = 390.

Therefore, the area of the square box is approximately 390 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 390 square feet is built; if each of the sides is √390, what will be the square feet of half of the building?

Okay, lets begin

195 square feet

Explanation

Since the building is square-shaped, we can simply divide the given area by 2.

Dividing 390 by 2 = 195 So, half of the building measures 195 square feet.

Well explained 👍

Problem 3

Calculate √390 x 5.

Okay, lets begin

Approximately 98.74

Explanation

First, find the square root of 390, which is approximately 19.74.

Then, multiply 19.74 by 5. So, 19.74 x 5 ≈ 98.74.

Well explained 👍

Problem 4

What will be the square root of (384 + 6)?

Okay, lets begin

The square root is 20.

Explanation

To find the square root, first find the sum of (384 + 6). 384 + 6 = 390, and then √390 ≈ 19.74.

Therefore, the square roots of (384 + 6) are ±19.74.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √390 units and the width ‘w’ is 40 units.

Okay, lets begin

The perimeter of the rectangle is approximately 119.48 units.

Explanation

Perimeter of the rectangle = 2 × (length + width)

Perimeter = 2 × (√390 + 40) ≈ 2 × (19.74 + 40) = 2 × 59.74 ≈ 119.48 units.

Well explained 👍

FAQ on Square Root of 390

1.What is √390 in its simplest form?

The prime factorization of 390 is 2 x 3 x 5 x 13, so the simplest form of √390 = √(2 x 3 x 5 x 13).

2.Mention the factors of 390.

The factors of 390 are 1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, and 390.

3.Calculate the square of 390.

The square of 390 is found by multiplying the number by itself: 390 x 390 = 152,100.

4.Is 390 a prime number?

5.390 is divisible by?

390 has many factors, including 1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, and 390.

Important Glossaries for the Square Root of 390

  • Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root, so √16 = 4.
     
  • Irrational number: An irrational number cannot be written as a simple fraction, where p and q are integers, and q is not equal to zero.
     
  • Principal square root: A number has both positive and negative square roots, but the positive square root is typically used in real-world applications, known as the principal square root.
     
  • Factorization: Breaking down a number into its basic components, or factors, that, when multiplied together, give the original number.
     
  • Approximation: Estimating a value that is close to the actual value, often used for numbers that cannot be expressed exactly, like irrational numbers.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.