130 LearnersLast updated on May 26th, 2025
Square Root of 1421
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 such as vehicle design, finance, etc. Here, we will discuss the square root of 1421.
What is the Square Root of 1421?
The square root is the inverse of the square of the number. 1421 is not a perfect square. The square root of 1421 is expressed in both radical and exponential form. In the radical form, it is expressed as √1421, whereas (1421)^(1/2) in the exponential form. √1421 ≈ 37.703, 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 1421
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 1421 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 1421 is broken down into its prime factors.
Step 1: Finding the prime factors of 1421 Breaking it down, we get 1421 = 13 x 109.
Step 2: Now we found out the prime factors of 1421. Since 1421 is not a perfect square, the digits of the number can't be grouped in pairs.
Therefore, calculating √1421 using prime factorization is not possible to get an exact whole number.
Square Root of 1421 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 1421, we group it as 21 and 14.
Step 2: Now we need to find n whose square is less than or equal to 14. We can choose n as '3' because 3 x 3 = 9 is less than 14. Now the quotient is 3, and the remainder is 14 - 9 = 5.
Step 3: Bring down the next pair of digits (21) to get 521 as the new dividend. Double the quotient (3) to get 6, which will be part of our new divisor.
Step 4: The new divisor will be in the form of 6n. We find n such that 6n x n is less than or equal to 521. Choosing n as 8, we have 68 x 8 = 544, which is greater than 521. Therefore, n should be 7, making 67 x 7 = 469.
Step 5: Subtract 469 from 521 to get a remainder of 52, and the quotient is 37.
Step 6: Since the dividend is less than the divisor, we need to add a decimal point and bring down two zeroes to make the new dividend 5200.
Step 7: Double the quotient (37) to get 74, which forms the next new divisor part. Repeat the process to find the next digit of the quotient.
Step 8: Continue doing these steps until you get the desired precision.
The square root of √1421 is approximately 37.703.
Square Root of 1421 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 1421 using the approximation method.
Step 1: Identify two perfect squares between which 1421 lies.
The smallest perfect square is 1369 (37^2) and the largest is 1444 (38^2). √1421 falls between 37 and 38.
Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (1421 - 1369) / (1444 - 1369) = 52 / 75 ≈ 0.693 Step 3: Add this result to the smaller square root: 37 + 0.693 = 37.693
Therefore, the square root of 1421 is approximately 37.693.
Common Mistakes and How to Avoid Them in the Square Root of 1421
Students often make mistakes while finding the square root, such as forgetting the negative square root or skipping steps in the long division method. Now let us look at a few common 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 principal (positive) square root, as it is the required one.
For example: √50 = 7.07, but 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 symbol properly is another mistake that students make. To resolve this, we need to teach students that simplifying the numbers inside the square root is important before proceeding to further calculations.
For Example: √(12 + 24) = √36, not √12 + √24.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to students 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 ≈ 7.071067, not 7.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube roots and square roots constantly as this will help them avoid mistakes.
For example: √50 and ∛50 are different.
Mistake 5
Making mistakes in the 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 1421 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √1421?
The area of the square is 1421 square units.
Explanation
The area of the square = side^2.
The side length is given as √1421.
Area of the square = side^2 = √1421 x √1421 = 1421.
Therefore, the area of the square box is 1421 square units.
Problem 2
A square-shaped building measuring 1421 square feet is built; if each of the sides is √1421, what will be the square feet of half of the building?
710.5 square feet.
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 1421 by 2 = we get 710.5.
So half of the building measures 710.5 square feet.
Problem 3
Calculate √1421 x 5.
188.515
Explanation
The first step is to find the square root of 1421, which is approximately 37.703.
The second step is to multiply 37.703 with 5.
So 37.703 x 5 = 188.515.
Problem 4
What will be the square root of (1421 + 79)?
The square root is 38.
Explanation
To find the square root, we need to find the sum of (1421 + 79). 1421 + 79 = 1500, and then √1500 ≈ 38.729.
Therefore, the square root of (1421 + 79) is approximately ±38.729.
Problem 5
Find the perimeter of the rectangle if its length 'l' is √1421 units and the width 'w' is 38 units.
We find the perimeter of the rectangle as 151.406 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√1421 + 38) = 2 × (37.703 + 38) ≈ 2 × 75.703 = 151.406 units.
FAQ on Square Root of 1421
1.What is √1421 in its simplest form?
2.Mention the factors of 1421.
3.Calculate the square of 1421.
4.Is 1421 a prime number?
5.1421 is divisible by?
6.How does learning Algebra help students in United Arab Emirates make better decisions in daily life?
7.How can cultural or local activities in United Arab Emirates support learning Algebra topics such as Square Root of 1421?
8.How do technology and digital tools in United Arab Emirates support learning Algebra and Square Root of 1421?
9.Does learning Algebra support future career opportunities for students in United Arab Emirates?
Important Glossaries for the Square Root of 1421
- 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 has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
- Prime factorization: The process of expressing a number as the product of its prime factors.
- Long division method: A method used to find the square roots of non-perfect squares by dividing and averaging to reach a more precise value.
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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
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