153 LearnersLast updated on May 26th, 2025
Square Root of 1729
If a number is multiplied by itself, 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 1729.
What is the Square Root of 1729?
The square root is the inverse of the square of a number. 1729 is not a perfect square. The square root of 1729 is expressed in both radical and exponential form. In radical form, it is expressed as √1729, whereas in exponential form it is (1729)^(1/2). √1729 ≈ 41.5633, 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 1729
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, 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 1729 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 1729 is broken down into its prime factors:
Step 1: Finding the prime factors of 1729 Breaking it down, we get 7 x 13 x 19.
Step 2: Now we found out the prime factors of 1729. The second step is to make pairs of those prime factors. Since 1729 is not a perfect square, the digits of the number cannot be grouped in pairs.
Therefore, calculating √1729 using prime factorization is not straightforward.
Square Root of 1729 by Long Division Method
The long division method is particularly used for non-perfect square numbers. In this method, we check the closest perfect square number to the given number. Let us now learn how to find the square root using the long division method, step by step:
Step 1: Group the numbers from right to left. In the case of 1729, group it as 17 and 29.
Step 2: Find a number whose square is less than or equal to 17. This number is 4, since 4 x 4 = 16. The quotient is 4, and the remainder is 1 (17 - 16).
Step 3: Bring down the next pair, 29, to get the new dividend, 129.
Step 4: Double the previous quotient (4) to get 8, which will be part of the new divisor.
Step 5: Find a number n such that 8n x n is less than or equal to 129. In this case, n is 1, since 81 x 1 = 81.
Step 6: Subtract 81 from 129 to get a remainder of 48, and the quotient is 41.
Step 7: Add decimal points to the quotient and bring down pairs of zeros.
Step 8: Repeat the process to find the next digit(s) after the decimal point.
The square root of 1729 is approximately 41.563.
Square Root of 1729 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 1729 using the approximation method.
Step 1: Find the closest perfect squares around 1729.
The smallest is 1600 (40^2) and the largest is 1764 (42^2).
√1729 falls somewhere between 40 and 42.
Step 2: Apply the formula:
(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).
(1729 - 1600) / (1764 - 1600) = 129 / 164 ≈ 0.786.
Step 3: Add this decimal to the smaller square root: 40 + 0.786 = 40.786.
The square root of 1729 is approximately 41.563, refining the approximation from the previous steps.
Common Mistakes and How to Avoid Them in the Square Root of 1729
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few of these mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to remember that a number has both positive and negative square roots. However, we typically consider only the positive square root, as it is the most useful in real-world applications.
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 correctly is another mistake. Teach students to simplify the numbers inside the square root first, then proceed.
For example, √(12 + 24) = √36, not √12 + √24.
Mistake 3
Not finding the correct value of a non-perfect square
The first step is to simplify the numbers inside the square root. This is crucial to avoid mistakes in identifying the approximate value.
For example, √50 ≈ 7.071067, not 7.
Mistake 4
Confusing the square root symbol with the cube root
Teach students the difference between cube roots and square roots to prevent confusion.
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, so students might accidentally skip a step, leading to a wrong answer. This may occur during subtraction or elsewhere in the process.
Square Root of 1729 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √1729?
The area of the square is 1729 square units.
Explanation
The area of the square = side^2. The side length is given as √1729. Area of the square = side^2 = √1729 x √1729 = 1729. Therefore, the area of the square box is 1729 square units.
Problem 2
A square-shaped building measuring 1729 square feet is built; if each of the sides is √1729, what will be the square feet of half of the building?
864.5 square feet
Explanation
We can divide the given area by 2 as the building is square-shaped. Dividing 1729 by 2, we get 864.5. So half of the building measures 864.5 square feet.
Problem 3
Calculate √1729 x 5.
207.8165
Explanation
The first step is to find the square root of 1729, which is approximately 41.563, and then multiply by 5. So, 41.563 x 5 ≈ 207.8165.
Problem 4
What will be the square root of (1600 + 129)?
The square root is approximately 41.563.
Explanation
To find the square root, find the sum of (1600 + 129) = 1729, then √1729 ≈ 41.563. Therefore, the square root of (1600 + 129) is approximately ±41.563.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √1729 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 159.126 units.
Explanation
Perimeter of the rectangle = 2 × (length + width) Perimeter = 2 × (√1729 + 38) ≈ 2 × (41.563 + 38) = 2 × 79.563 ≈ 159.126 units.
FAQ on Square Root of 1729
1.What is √1729 in its simplest form?
2.Mention the factors of 1729.
3.Calculate the square of 1729.
4.Is 1729 a prime number?
5.1729 is divisible by?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 1729?
8.How do technology and digital tools in India support learning Algebra and Square Root of 1729?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 1729
- Square root: A square root is the inverse operation to squaring a number. Example: 4^2 = 16, and the inverse of squaring is the square root, so √16 = 4.
- Irrational number: An irrational number is a number that cannot be written as a simple fraction (p/q), where p and q are integers and q is not zero.
- Approximation: The process of finding a value that is close enough to the right answer, typically with some thought or calculation involved.
- Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, and 16 are perfect squares.
- Long division method: A method used to find the square root of a number by performing division and averaging processes iteratively.
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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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