135 LearnersLast updated on May 26th, 2025
Square Root of 1400
If a number is multiplied by itself, the result is a square. The inverse of squaring a number is finding its square root. Square roots are used in fields such as vehicle design, finance, and engineering. Here, we will discuss the square root of 1400.
What is the Square Root of 1400?
The square root is the inverse operation of squaring a number. 1400 is not a perfect square. The square root of 1400 can be expressed in both radical and exponential forms. In radical form, it is expressed as √1400, whereas in exponential form, it is written as (1400)^(1/2). √1400 ≈ 37.41657, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 1400
The prime factorization method is useful for perfect squares. However, for non-perfect squares, the long division method and approximation method are used. Let's explore these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 1400 by Prime Factorization Method
The prime factorization of a number is the product of prime factors. Let's break down 1400 into its prime factors:
Step 1: Find the prime factors of 1400. Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 7, which is 2^3 x 5^2 x 7.
Step 2: Now, pair the prime factors. Since 1400 is not a perfect square, the digits of the number can’t be grouped into complete pairs.
Therefore, calculating √1400 using prime factorization gives an approximate answer.
Square Root of 1400 by Long Division Method
The long division method is used for non-perfect squares. Let's find the square root using this method step by step:
Step 1: Group the numbers from right to left. For 1400, group it as 14 and 00.
Step 2: Find a number whose square is less than or equal to 14. This number is 3, as 3 x 3 = 9. Subtract 9 from 14 to get a remainder of 5.
Step 3: Bring down the next pair of digits, 00, to make the new dividend 500.
Step 4: Double the previous quotient (3), resulting in 6, and determine a digit n such that 6n x n ≤ 500.
Step 5: Trying n = 8, we calculate 68 x 8 = 544, which is slightly more than 500. Try n = 7, which gives 67 x 7 = 469.
Step 6: Subtract 469 from 500 to get a remainder of 31. The quotient is now 37.
Step 7: Add a decimal point to the quotient and bring down two zeros to make the new dividend 3100. Step 8: Continue the process until you achieve the desired decimal places.
The square root of 1400 is approximately 37.41657.
Square Root of 1400 by Approximation Method
The approximation method provides an easy way to estimate the square root of a number. Here's how to find the square root of 1400 using this method:
Step 1: Identify the nearest perfect squares around 1400.
The closest perfect square less than 1400 is 1369, and the closest perfect square greater than 1400 is 1444. Therefore, √1400 is between 37 and 38.
Step 2: Use the formula (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) to find the decimal part. Step 3: (1400 - 1369) / (1444 - 1369) = 31 / 75 ≈ 0.413 Add this decimal to the whole number part to get 37 + 0.413 = 37.413. So, the approximate square root of 1400 is 37.413.
Common Mistakes and How to Avoid Them in the Square Root of 1400
Students often make mistakes when finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's explore some common mistakes in detail.
Mistake 1
Forgetting about the Negative Square Root
It's important to remember that every positive number has both positive and negative square roots. However, we typically consider only the positive square root in practical applications.
For example: √50 = 7.07, but there is also -7.07, which should not be ignored conceptually.
Mistake 2
Not Adding the Square Root Symbol in Proper Places
A common mistake is misplacing the square root symbol. It is important to simplify expressions inside the square root before proceeding.
For example: √(16 + 9) = √25, not √16 + √9.
Mistake 3
Not Finding the Correct Value of a Non-Perfect Square
Students often simplify incorrectly when dealing with non-perfect squares. To avoid mistakes, ensure the correct identification of approximate values.
For example: √50 ≈ 7.071067, not 7.
Mistake 4
Confusing the Square Root Symbol with the Cube Root
Students must distinguish between square roots and cube roots.
For example: √50 and ∛50 are different operations.
Mistake 5
Making Mistakes in Long Division
The long division method involves numerous steps, and skipping a step can lead to incorrect answers. This may occur in subtraction or other parts of the process.
Square Root of 1400 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √1400?
The area of the square is approximately 1400 square units.
Explanation
The area of the square is calculated as side^2.
The side length is given as √1400.
Area of the square = (√1400) × (√1400) = 1400
Therefore, the area of the square box is approximately 1400 square units.
Problem 2
A square-shaped building measuring 1400 square feet is built; if each side is √1400, what will be the square feet of half of the building?
700 square feet
Explanation
We can divide the given area by 2 as the building is square-shaped.
Dividing 1400 by 2, we get 700.
So half of the building measures 700 square feet.
Problem 3
Calculate √1400 × 5.
187.08
Explanation
The first step is to find the square root of 1400, which is approximately 37.41657.
The second step is to multiply 37.41657 by 5.
So, 37.41657 × 5 ≈ 187.08.
Problem 4
What is the square root of (1400 + 100)?
The square root is 40.
Explanation
To find the square root, first calculate the sum of (1400 + 100). 1400 + 100 = 1500, and then √1500 ≈ 38.72983.
For simplicity, rounding might suggest approximate solutions, so using nearby perfect squares for better approximation is advised.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √1400 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as 154.83 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√1400 + 40) ≈ 2 × (37.41657 + 40) ≈ 2 × 77.41657 ≈ 154.83 units.
FAQ on Square Root of 1400
1.What is √1400 in its simplest form?
2.Mention the factors of 1400.
3.Calculate the square of 1400.
4.Is 1400 a prime number?
5.1400 is divisible by?
6.How does learning Algebra help students in Thailand make better decisions in daily life?
7.How can cultural or local activities in Thailand support learning Algebra topics such as Square Root of 1400?
8.How do technology and digital tools in Thailand support learning Algebra and Square Root of 1400?
9.Does learning Algebra support future career opportunities for students in Thailand?
Important Glossaries for the Square Root of 1400
- Square root: A square root is the inverse of a square. For example: 4^2 = 16, and the inverse of the square is the square root, which is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written as a simple fraction; its decimal representation is non-terminating and non-repeating.
- Principal square root: A number has both positive and negative square roots; however, the positive square root is often used in real-world applications.
- Prime factorization: Breaking down a number into the product of its prime factors.
- Decimal: A number that includes a whole part and a fractional part, represented with a decimal point, such as 37.41657.
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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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