103 LearnersLast updated on May 26th, 2025
Square Root of 1469
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 like vehicle design, finance, etc. Here, we will discuss the square root of 1469.
What is the Square Root of 1469?
The square root is the inverse of the square of the number. 1469 is not a perfect square. The square root of 1469 is expressed in both radical and exponential form. In the radical form, it is expressed as √1469, whereas (1469)^(1/2) in the exponential form. √1469 ≈ 38.3319, 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 1469
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division and approximation methods are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 1469 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 1469 is broken down into its prime factors:
Step 1: Finding the prime factors of 1469 Breaking it down, we get 7 x 7 x 3 x 31: 7^2 x 3 x 31
Step 2: Now we found out the prime factors of 1469. The second step is to make pairs of those prime factors. Since 1469 is not a perfect square, the digits of the number can’t be grouped in complete pairs.
Therefore, calculating √1469 using prime factorization is not straightforward.
Square Root of 1469 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 1469, we need to group it as 69 and 14.
Step 2: Now we need to find n whose square is 14 or less. We can say n is 3 because 3 x 3 = 9 is less than 14. Now the quotient is 3, and after subtracting 9 from 14, the remainder is 5.
Step 3: Now let us bring down 69, making the new dividend 569. Add the old divisor with the same number, 3 + 3, to get 6, which will be our new divisor.
Step 4: The new divisor will be 6n. We need to find the value of n such that 6n x n ≤ 569. Let us consider n as 9, now 69 x 9 = 621.
Step 5: Since 621 is greater than 569, try n as 8. So, 68 x 8 = 544.
Step 6: Subtract 544 from 569; the difference is 25. The quotient is now 38.
Step 7: Since the remainder is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2500.
Step 8: The next divisor will be 760 (from 76 with a 0 added) to find n such that 760n x n ≤ 2500. Let n be 3, then 760 x 3 = 2280.
Step 9: Subtracting 2280 from 2500 gives 220.
Step 10: Continue this process until we get two numbers after the decimal point.
So, the square root of 1469 ≈ 38.33
Square Root of 1469 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 1469 using the approximation method.
Step 1: Find the closest perfect squares to √1469.
The smallest perfect square less than 1469 is 1444 (38^2) and the largest perfect square more than 1469 is 1521 (39^2). √1469 falls between 38 and 39.
Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (1469 - 1444) / (1521 - 1444) = 25 / 77 ≈ 0.324
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 38 + 0.33 ≈ 38.33, so the square root of 1469 is approximately 38.33.
Common Mistakes and How to Avoid Them in the Square Root of 1469
Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now 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.
Square root of 1469 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √1469?
The area of the square is 1469 square units.
Explanation
The area of the square = side^2.
The side length is given as √1469.
Area of the square = side^2 = √1469 x √1469 = 1469.
Therefore, the area of the square box is 1469 square units.
Problem 2
A square-shaped building measuring 1469 square feet is built; if each of the sides is √1469, what will be the square feet of half of the building?
734.5 square feet
Explanation
To find half the building area, divide the given area by 2 since the building is square-shaped.
Dividing 1469 by 2 gives us 734.5.
So, half of the building measures 734.5 square feet.
Problem 3
Calculate √1469 x 5.
191.66
Explanation
The first step is to find the square root of 1469, which is approximately 38.33.
The second step is to multiply 38.33 by 5.
So, 38.33 x 5 ≈ 191.66.
Problem 4
What will be the square root of (1444 + 25)?
The square root is 39.
Explanation
To find the square root, we need to find the sum of (1444 + 25). 1444 + 25 = 1469, and then √1469 ≈ 38.33, but the original question should use perfect squares.
Instead, if it were √(1444 + 25) as perfect squares: √1469, the closest perfect square is 1521, which is 39.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √1469 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as 152.66 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√1469 + 38) = 2 × (38.33 + 38) = 2 × 76.33 = 152.66 units.
FAQ on Square Root of 1469
1.What is √1469 in its simplest form?
2.Mention the factors of 1469.
3.Calculate the square of 1469.
4.Is 1469 a prime number?
5.1469 is divisible by?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 1469?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 1469?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of 1469
- 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, the positive square root is often used in practical applications. That is the reason it is also known as the principal square root.
- Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.
- Long division method: A technique used to find the square root of a non-perfect square number, involving a step-by-step division process.
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Jaskaran Singh Saluja
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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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