Last updated on May 26th, 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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 4194.
The square root is the inverse of the square of the number. 4194 is not a perfect square. The square root of 4194 is expressed in both radical and exponential form. In the radical form, it is expressed as √4194, whereas (4194)^(1/2) in the exponential form. √4194 ≈ 64.7542, 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.
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:
The product of prime factors is the prime factorization of a number. Now let us look at how 4194 is broken down into its prime factors:
Step 1: Finding the prime factors of 4194 Breaking it down, we get 2 x 3 x 11 x 11 x 7: 2^1 x 3^1 x 11^2 x 7^1
Step 2: Now we found out the prime factors of 4194. The second step is to make pairs of those prime factors. Since 4194 is not a perfect square, calculating 4194 using prime factorization involves taking the square root of the product of any pairs of factors.
The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:
Step 1: To begin with, group the numbers from right to left in pairs. In the case of 4194, we group it as 41 and 94.
Step 2: Now find a number whose square is less than or equal to 41. We can say '6' because 6 x 6 = 36, which is less than 41. Now the quotient is 6, and the remainder is 5.
Step 3: Bring down 94 to make the new dividend 594. Double the quotient and place it next to the divisor, making it 12.
Step 4: Find a digit 'n' such that 12n × n ≤ 594. Let's try '4'; 124 x 4 = 496.
Step 5: Subtract 496 from 594; the remainder is 98. The quotient is now 64.
Step 6: Since the dividend is less than the divisor, add a decimal point and two zeros. The new dividend is 9800.
Step 7: Continue with the long division to obtain the next digits after the decimal until the desired precision is reached.
So, the square root of √4194 ≈ 64.75.
The approximation method is another method for finding the 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 4194 using the approximation method.
Step 1: Find the closest perfect squares to 4194. The smallest perfect square less than 4194 is 4096 (64^2), and the largest perfect square greater than 4194 is 4225 (65^2). So, √4194 is between 64 and 65.
Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (4194 - 4096) ÷ (4225 - 4096) = 98 / 129 ≈ 0.76 Adding this to 64, we get 64 + 0.76 = 64.76, so the square root of 4194 is approximately 64.76.
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √4194?
The area of the square is 4194 square units.
The area of the square = side^2.
The side length is given as √4194.
Area of the square = side^2 = √4194 × √4194 = 4194.
Therefore, the area of the square box is 4194 square units.
A square-shaped building measuring 4194 square feet is built; if each of the sides is √4194, what will be the square feet of half of the building?
2097 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 4194 by 2, we get 2097.
So, half of the building measures 2097 square feet.
Calculate √4194 × 5.
323.77
The first step is to find the square root of 4194, which is approximately 64.75.
The second step is to multiply 64.75 by 5.
So, 64.75 × 5 = 323.75.
What will be the square root of (4096 + 98)?
The square root is 65.
To find the square root, calculate the sum of (4096 + 98).
4096 + 98 = 4194, and then √4194 ≈ 64.75.
Therefore, the square root of (4096 + 98) is approximately 64.75.
Find the perimeter of the rectangle if its length ‘l’ is √4194 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as 209.51 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√4194 + 40)
= 2 × (64.75 + 40)
≈ 2 × 104.75
= 209.5 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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