124 LearnersLast updated on May 26th, 2025
Square Root of 1378
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 1378.
What is the Square Root of 1378?
The square root is the inverse of the square of the number. 1378 is not a perfect square. The square root of 1378 is expressed in both radical and exponential form. In radical form, it is expressed as √1378, whereas (1378)^(1/2) in the exponential form. √1378 ≈ 37.122, 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 1378
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:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 1378 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 1378 is broken down into its prime factors.
Step 1: Finding the prime factors of 1378 1378 can be broken down as 2 x 689. Further breaking down 689, we get 689 = 13 x 53. Therefore, 1378 = 2 x 13 x 53.
Step 2: Now we found out the prime factors of 1378. The second step is to make pairs of those prime factors. Since 1378 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √1378 using prime factorization does not yield an exact value.
Square Root of 1378 by Long Division Method
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, we need to group the numbers from right to left. In the case of 1378, we need to group it as 78 and 13.
Step 2: Now we need to find n whose square is 13. We can say n is ‘3’ because 3^2 = 9 is less than 13. Now the quotient is 3. Subtracting 9 from 13, the remainder is 4.
Step 3: Now let us bring down 78, which is the new dividend. Add the old divisor with the same number: 3 + 3 = 6, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor; we need to find the value of n.
Step 5: The next step is finding 6n x n ≤ 478. Let us consider n as 7. Now 67 x 7 = 469.
Step 6: Subtract 469 from 478; the difference is 9, and the quotient is 37.
Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 900.
Step 8: Now we need to find the new divisor, which is 371 because 371 x 2 = 742.
Step 9: Subtracting 742 from 900, we get the result 158.
Step 10: Now the quotient is 37.12
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values; continue till the remainder is zero.
So the square root of √1378 is approximately 37.12.
Square Root of 1378 by Approximation Method
The approximation method 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 1378 using the approximation method.
Step 1: Now we have to find the closest perfect square of √1378. The smallest perfect square less than 1378 is 1369 (37^2), and the largest perfect square greater than 1378 is 1444 (38^2). √1378 falls somewhere between 37 and 38.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1378 - 1369) ÷ (1444 - 1369) ≈ 0.12. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 37 + 0.12 = 37.12, so the square root of 1378 is approximately 37.12.
Common Mistakes and How to Avoid Them in the Square Root of 1378
Students make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Let us look at a few of those mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example: √50 = 7.07, there is also -7.07 which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the second step.
For Example: √(12+24) = √12 + √24 is the wrong method of using roots.
The right way of using roots is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For Example: √50 = 7 is wrong; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between the cube root and the square root constantly, as this will help them avoid mistakes.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, which results in a wrong answer at the end of the sum. This may either happen in subtracting or elsewhere.
Square Root of 1378 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √1398?
The area of the square is approximately 1398 square units.
Explanation
The area of the square = side^2.
The side length is given as √1398.
Area of the square = side^2 = √1398 x √1398 = 1398.
Therefore, the area of the square box is 1398 square units.
Problem 2
A square-shaped garden measuring 1378 square feet is built; if each of the sides is √1378, what will be the square feet of half of the garden?
689 square feet
Explanation
We can just divide the given area by 2 as the garden is square-shaped.
Dividing 1378 by 2, we get 689.
So half of the garden measures 689 square feet.
Problem 3
Calculate √1378 x 4.
148.488
Explanation
The first step is to find the square root of 1378, which is approximately 37.122.
The second step is to multiply 37.122 by 4.
So 37.122 x 4 ≈ 148.488.
Problem 4
What will be the square root of (1378 + 22)?
The square root is approximately 38.
Explanation
To find the square root, we need to find the sum of (1378 + 22). 1378 + 22 = 1400, and then √1400 ≈ 37.416.
Therefore, the square root of (1378 + 22) is approximately 37.416.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √1378 units and the width ‘w’ is 50 units.
We find the perimeter of the rectangle as approximately 174.244 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√1378 + 50) = 2 × (37.122 + 50) ≈ 2 × 87.122 ≈ 174.244 units.
FAQ on Square Root of 1378
1.What is √1378 in its simplest form?
2.What are the factors of 1378?
3.Calculate the square of 1378.
4.Is 1378 a prime number?
5.1378 is divisible by?
6.How does learning Algebra help students in Bahrain make better decisions in daily life?
7.How can cultural or local activities in Bahrain support learning Algebra topics such as Square Root of 1378?
8.How do technology and digital tools in Bahrain support learning Algebra and Square Root of 1378?
9.Does learning Algebra support future career opportunities for students in Bahrain?
Important Glossaries for the Square Root of 1378
- Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
- Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is most commonly used due to its applications in the real world. That is why it is called the principal square root.
- Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
- Prime factorization: The process of dividing a number into its basic prime factors. For example, the prime factorization of 1378 is 2 x 13 x 53.
Explore More algebra
About BrightChamps in Bahrain
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.
