139 LearnersLast updated on May 26th, 2025
Square Root of 735
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 735.
What is the Square Root of 735?
The square root is the inverse of the square of the number. 735 is not a perfect square. The square root of 735 is expressed in both radical and exponential form. In the radical form, it is expressed as √735, whereas (735)^(1/2) in the exponential form. √735 ≈ 27.114, 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 735
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 735 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 735 is broken down into its prime factors.
Step 1: Finding the prime factors of 735
Breaking it down, we get 3 × 5 × 7 × 7: 3^1 × 5^1 × 7^2
Step 2: Now we found out the prime factors of 735. The second step is to make pairs of those prime factors. Since 735 is not a perfect square, the digits of the number can’t be grouped in pairs to completely factor it as a square. Therefore, calculating the square root of 735 using prime factorization alone is not straightforward.
Square Root of 735 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 735, we need to group it as 35 and 7.
Step 2: Now we need to find n whose square is less than or equal to 7. We can say n is ‘2’ because 2 × 2 = 4, which is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.
Step 3: Bring down 35, making the new dividend 335. Double the quotient 2 to get 4, which will be part of our new divisor.
Step 4: We need to find a number n such that 4n × n ≤ 335. Let us consider n as 8, now 48 × 8 = 384, which is greater than 335. Try n = 7, then 47 × 7 = 329.
Step 5: Subtracting 329 from 335, we get the remainder 6.
Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros, making the new dividend 600.
Step 7: The new divisor is 547, because 547 × 1 = 547
Step 8: Subtracting 547 from 600 gives 53.
Step 9: Continue doing these steps until we get two numbers after the decimal point.
So the square root of √735 ≈ 27.114
Square Root of 735 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 735 using the approximation method.
Step 1: Now we have to find the closest perfect square of √735 The smallest perfect square less than 735 is 729 and the closest larger perfect square is 784. √735 falls between 27 and 28.
Step 2: Now we need to apply the formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square) Going by the formula (735 - 729) ÷ (784 - 729) ≈ 0.109 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 27 + 0.109 = 27.109, so the square root of 735 is approximately 27.109
Common Mistakes and How to Avoid Them in the Square Root of 735
Students sometimes make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. 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 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 forgotten.
Mistake 2
Not adding square root symbols in proper places
Not using the square root 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.
For example, √(12+24) = √12 + √24 is incorrect.
The correct way is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
It is crucial to teach students to approximate the values of non-perfect squares accurately.
For example, √50 ≈ 7.071067, not simply 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 to avoid 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. Due to this, many students might skip a step accidentally, resulting in a wrong answer. This may happen during subtraction or elsewhere.
Square root of 735 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √735?
The area of the square is approximately 540.225 square units.
Explanation
The area of the square is calculated as side^2.
The side length is given as √735.
Area of the square = side^2 = √735 × √735 ≈ 27.114 × 27.114 ≈ 735.
Therefore, the area of the square box is approximately 735 square units.
Problem 2
A square-shaped building measuring 735 square feet is built; if each of the sides is √735, what will be the square feet of half of the building?
367.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 735 by 2, we get 367.5.
So half of the building measures 367.5 square feet.
Problem 3
Calculate √735 × 5.
135.57
Explanation
The first step is to find the square root of 735, which is approximately 27.114.
The second step is to multiply 27.114 by 5.
So 27.114 × 5 ≈ 135.57.
Problem 4
What will be the square root of (729 + 6)?
The square root is approximately 27.185.
Explanation
To find the square root, we need to find the sum of (729 + 6). 729 + 6 = 735, and then √735 ≈ 27.114.
Therefore, the square root of (729 + 6) is approximately ±27.114.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √735 units and the width ‘w’ is 35 units.
The perimeter of the rectangle is approximately 124.228 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√735 + 35) ≈ 2 × (27.114 + 35) ≈ 2 × 62.114 ≈ 124.228 units.
FAQ on Square Root of 735
1.What is √735 in its simplest form?
2.Mention the factors of 735.
3.Calculate the square of 735.
4.Is 735 a prime number?
5.735 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 735?
8.How do technology and digital tools in India support learning Algebra and Square Root of 735?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 735
- Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16 and the inverse operation is the square root, that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written as a simple fraction, where the denominator is not zero. For example, √2 is irrational.
- Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 9 is 3.
- Prime Factorization: The expression of a number as a product of its prime factors. For example, the prime factorization of 735 is 3 × 5 × 7 × 7.
- Decimal: A decimal number includes a whole number and a fractional part separated by a decimal point, such as 7.86.
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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.
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