Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. Square roots are used in various fields such as engineering, finance, etc. Here, we will discuss the square root of 3145.
The square root is the inverse of squaring a number. 3145 is not a perfect square. The square root of 3145 is expressed in both radical and exponential form. In the radical form, it is expressed as √3145, whereas in exponential form as (3145)^(1/2). √3145 ≈ 56.072, which is an irrational number because it cannot be expressed as a fraction of two integers.
For perfect square numbers, the prime factorization method is commonly used. However, for non-perfect square numbers like 3145, methods such as the long division method and approximation method are utilized. Let us now explore these methods:
The product of prime factors is the prime factorization of a number. Let's break down 3145 into its prime factors:
Step 1: Finding the prime factors of 3145 Breaking it down, we get 5 x 13 x 29 x 1: 5^1 x 13^1 x 29^1
Step 2: Since 3145 is not a perfect square, the digits of the number cannot be grouped into pairs. Therefore, calculating 3145 using prime factorization is impractical.
The long division method is particularly useful for non-perfect square numbers. Here is how to find the square root using the long division method, step by step:
Step 1: Group the digits of 3145 starting from the right. So, we have 31 and 45.
Step 2: Find the largest number whose square is less than or equal to 31. This number is 5, as 5^2 = 25.
Step 3: Subtract 25 from 31, leaving a remainder of 6. Bring down the next pair, 45, to make the new dividend 645.
Step 4: Double the previous quotient (5) to get 10, which becomes part of the new divisor.
Step 5: Find a digit (n) such that 10n * n gives a product less than or equal to 645. The value of n is 6.
Step 6: Subtract the product, 636, from 645, leaving a remainder of 9. Bring down two zeros to continue.
Step 7: Continue this process to calculate a precise decimal value. The quotient grows to reflect the square root. Thus, the square root of 3145 is approximately 56.072.
The approximation method is another way to find square roots. Here's how to find the square root of 3145 using this method:
Step 1: Identify the closest perfect squares around 3145. The nearest perfect squares are 3136 (56^2) and 3249 (57^2). So, √3145 falls between 56 and 57.
Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Applying the formula: (3145 - 3136) / (3249 - 3136) ≈ 0.081 Adding this decimal to the lower square root gives 56 + 0.081 = 56.081. However, upon more precise calculation, the square root of 3145 is approximately 56.072.
Students often make mistakes when finding square roots, such as forgetting about the negative square root or misapplying methods. Let's address a few common errors:
Can you help Max find the area of a square box if its side length is given as √3145?
The area of the square is approximately 3145 square units.
The area of a square is calculated as side^2.
Given the side length as √3145, the area is:
Area = (√3145) * (√3145) = 3145 square units.
A square-shaped building measuring 3145 square feet is built; if each side is √3145, what will be the square feet of half of the building?
1572.5 square feet
Divide the total area by 2 to find half of the building's area: 3145 / 2 = 1572.5 square feet.
Calculate √3145 × 10.
Approximately 560.72
First, find the square root of 3145, which is approximately 56.072, then multiply by 10: 56.072 × 10 = 560.72.
What will be the square root of (3140 + 5)?
The square root is approximately 56.072
To find the square root, sum 3140 and 5 to get 3145, then calculate the square root: √3145 ≈ 56.072.
Find the perimeter of the rectangle if its length ‘l’ is √3145 units and the width ‘w’ is 50 units.
The perimeter of the rectangle is approximately 212.144 units.
Perimeter of the rectangle = 2 × (length + width) Perimeter = 2 × (√3145 + 50) ≈ 2 × (56.072 + 50) = 2 × 106.072 = 212.144 units.
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.
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