120 LearnersLast updated on May 26th, 2025
Square Root of 734
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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 734.
What is the Square Root of 734?
The square root is the inverse of the square of a number. 734 is not a perfect square. The square root of 734 is expressed in both radical and exponential form. In radical form, it is expressed as √734, whereas in exponential form it is (734)^(1/2). √734 ≈ 27.086, 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 734
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 734 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 734 is broken down into its prime factors:
Step 1: Finding the prime factors of 734
Breaking it down, we get 2 x 367: 2^1 x 367^1
Step 2: Now we found out the prime factors of 734. The second step is to make pairs of those prime factors. Since 734 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 734 using prime factorization is not straightforward.
Square Root of 734 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 734, we need to group it as 34 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. After subtracting 4 from 7, the remainder is 3.
Step 3: Now bring down 34, making the new dividend 334. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.
Step 4: The new divisor will be 4n. We need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 334. Let us consider n as 7, now 4 × 7 × 7 = 329.
Step 6: Subtract 329 from 334, the difference is 5, and the quotient is 27.
Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 500.
Step 8: Now we need to find the new divisor that is 541 because 541 × 1 = 541.
Step 9: Subtracting 541 from 500 is not possible, so consider 27.0 and continue the process until you have two decimal places.
The square root of √734 is approximately 27.086.
Square Root of 734 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 734 using the approximation method.
Step 1: Now we have to find the closest perfect squares to √734. The smallest perfect square less than 734 is 729, and the largest perfect square greater than 734 is 784. √734 falls somewhere between 27 and 28.
Step 2: Now apply the formula: Given number - smallest perfect square / (Greater perfect square - smallest perfect square) Going by the formula (734 - 729) ÷ (784 - 729) ≈ 0.086. 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: 27 + 0.086 ≈ 27.086. So, the square root of 734 is approximately 27.086.
Common Mistakes and How to Avoid Them in the Square Root of 734
Students often make mistakes while finding the square root, such as forgetting about the negative square root and skipping long division steps. Let's look at a few mistakes 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 use only the positive square root, as it is the most relevant in many 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 properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important before moving on to the next step.
For example: √(12 + 24) = √12 + √24 is incorrect.
The correct approach is √(12 + 24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children 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 is incorrect; the correct answer is √50 ≈ 7.071067.
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 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 at the end of the process. This may occur in subtraction or elsewhere in the steps.
Square root of 734 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √734?
The area of the square is approximately 538.324 square units.
Explanation
The area of the square = side^2.
The side length is given as √734.
Area of the square = side^2 = √734 × √734 ≈ 27.086 × 27.086 ≈ 734
Therefore, the area of the square box is approximately 538.324 square units.
Problem 2
A square-shaped building measuring 734 square feet is built; if each of the sides is √734, what will be the square feet of half of the building?
367 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 734 by 2, we get 367.
So half of the building measures 367 square feet.
Problem 3
Calculate √734 × 5.
Approximately 135.43
Explanation
The first step is to find the square root of 734, which is approximately 27.086.
The second step is to multiply 27.086 by 5.
So 27.086 × 5 ≈ 135.43
Problem 4
What will be the square root of (734 + 6)?
The square root is approximately 24.58
Explanation
To find the square root, we need to find the sum of (734 + 6). 734 + 6 = 740, and then √740 ≈ 27.195.
Therefore, the square root of (734 + 6) is approximately 27.195.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √734 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 130.172 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√734 + 38) ≈ 2 × (27.086 + 38) ≈ 2 × 65.086 ≈ 130.172 units.
FAQ on Square Root of 734
1.What is √734 in its simplest form?
2.Mention the factors of 734.
3.Calculate the square of 734.
4.Is 734 a prime number?
5.734 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 734?
8.How do technology and digital tools in India support learning Algebra and Square Root of 734?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 734
- 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 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 more in real-world applications. This is known as the principal square root.
- Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.
- Prime factorization: Prime factorization is the process of expressing a number as the product of prime numbers.
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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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