Square Root of 1045
2026-02-28 11:49 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields like engineering, finance, etc. Here, we will discuss the square root of 1045.

What is the Square Root of 1045?

The square root is the inverse of the square of the number. 1045 is not a perfect square. The square root of 1045 is expressed in both radical and exponential form. In the radical form, it is expressed as √1045, whereas (1045)^(1/2) in exponential form. √1045 ≈ 32.343, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 1045

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long division and approximation methods are used. Let us now learn the following methods: 

  • Prime factorization method 
  • Long division method 
  • Approximation method

Square Root of 1045 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 1045 is broken down into its prime factors.

Step 1: Finding the prime factors of 1045 Breaking it down, we get 5 × 11 × 19.

Step 2: We found the prime factors of 1045. Since 1045 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √1045 using prime factorization directly is not possible.

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Square Root of 1045 by Long Division Method

The long division method is particularly used for non-perfect square numbers. Let us now 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 1045, we need to group it as 10 and 45.

Step 2: Find a number whose square is less than or equal to 10. We can say this number is ‘3’ because 3 × 3 = 9. The quotient is 3, and the remainder is 1 after subtracting 9 from 10.

Step 3: Bring down the next pair, which is 45, making the new dividend 145. Add the old divisor with the same number, 3 + 3 = 6, to get the new divisor.

Step 4: The new divisor will be 6n. Find the largest value of n such that 6n × n ≤ 145. The value n is 2, so 62 × 2 = 124.

Step 5: Subtract 124 from 145, the result is 21, and the quotient is 32.

Step 6: Since the dividend is less than the divisor, add a decimal point. Add two zeroes to the dividend to make it 2100.

Step 7: The new divisor is 644. Find a number n such that 644n × n is less than or equal to 2100. Continue this process to achieve better precision.

So, the square root of √1045 is approximately 32.343.

Square Root of 1045 by Approximation Method

Approximation is another method for finding square roots. It is an easy method to find the square root of a given number. Now, let us learn how to find the square root of 1045 using the approximation method.

Step 1: Find the closest perfect squares of √1045. The smallest perfect square less than 1045 is 1024 (32^2), and the largest perfect square greater than 1045 is 1089 (33^2). √1045 falls between 32 and 33.

Step 2: Apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using (1045 - 1024) / (1089 - 1024) = 21 / 65 = 0.323.

The approximate square root is 32 + 0.323 = 32.323.

Common Mistakes and How to Avoid Them in the Square Root of 1045

Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping the long division method. Let us look at a few 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 √1045?

Okay, lets begin

The area of the square is approximately 1045 square units.

Explanation

The area of a square = side^2.

The side length is given as √1045.

Area = (√1045) × (√1045) = 1045.

Well explained 👍

Problem 2

A square-shaped building measuring 1045 square feet is built; if each side is √1045, what will be the square feet of half of the building?

Okay, lets begin

522.5 square feet

Explanation

Divide the total area by 2 since the building is square-shaped.

1045 / 2 = 522.5 square feet.

Well explained 👍

Problem 3

Calculate √1045 × 3.

Okay, lets begin

Approximately 97.029

Explanation

First, find the square root of 1045, which is approximately 32.343. Then multiply 32.343 by 3: 32.343 × 3 ≈ 97.029.

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Problem 4

What will be the square root of (1024 + 21)?

Okay, lets begin

The square root is 33.

Explanation

Find the sum of (1024 + 21) = 1045, then find the square root of 1045, which is approximately 32.343, but since this is a sum leading to a perfect square (1089), it simplifies to 33.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √1045 units and the width ‘w’ is 25 units.

Okay, lets begin

The perimeter of the rectangle is approximately 114.686 units.

Explanation

Perimeter of a rectangle = 2 × (length + width).

Perimeter = 2 × (√1045 + 25)

= 2 × (32.343 + 25)

≈ 114.686 units.

Well explained 👍

FAQ on Square Root of 1045

1.What is √1045 in its simplest form?

The prime factorization of 1045 is 5 × 11 × 19, so the simplest form of √1045 cannot be further simplified into a perfect square factor form.

2.Mention the factors of 1045.

Factors of 1045 are 1, 5, 11, 19, 55, 95, 209, and 1045.

3.Calculate the square of 1045.

The square of 1045 is 1045 × 1045 = 1,092,025.

4.Is 1045 a prime number?

1045 is not a prime number, as it has more than two factors.

5.1045 is divisible by?

1045 is divisible by 1, 5, 11, 19, 55, 95, 209, and 1045.

Important Glossaries for the Square Root of 1045

  • Square root: A square root is the inverse operation of squaring a number. For example, if 4^2 = 16, then √16 = 4.
     
  • Irrational number: An irrational number cannot be expressed as a simple fraction; it has a non-repeating, non-terminating decimal expansion.
     
  • Principal square root: The non-negative square root of a number. For example, the principal square root of 25 is 5.
     
  • Factors: Numbers you can multiply together to get another number. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
     
  • Perfect square: A number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.

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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.