129 LearnersLast updated on May 26th, 2025
Square Root of 749
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 fields of vehicle design, finance, etc. Here, we will discuss the square root of 749.
What is the Square Root of 749?
The square root is the inverse of the square of the number. 749 is not a perfect square. The square root of 749 is expressed in both radical and exponential form. In the radical form, it is expressed as √749, whereas (749)¹/₂ in the exponential form. √749 ≈ 27.370, 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 749
The prime factorization method is used for perfect square numbers. However, it 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 749 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 749 is broken down into its prime factors.
Step 1: Finding the prime factors of 749 Breaking it down, we get 7 x 107, where both 7 and 107 are prime numbers.
Step 2: Since 749 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 749 using prime factorization is not feasible for finding the square root.
Square Root of 749 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 749, we need to group it as 49 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 x 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 49, making the new dividend 349. Add the old divisor (2) with the same number (2) to get 4, which will be our new divisor.
Step 4: The new divisor will be 4n. Now, find n such that 4n x n ≤ 349. Let us consider n as 8, now 48 x 8 = 384.
Step 5: Since 349 is less than 384, use a smaller n. Try n = 7, 47 x 7 = 329.
Step 6: Subtract 329 from 349, the difference is 20, 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 zeroes to the dividend. Now, the new dividend is 2000.
Step 8: Find the new divisor by adding 7 to 47, making it 54. Find n such that 54n x n ≤ 2000.
Step 9: Continue the process until a satisfactory precision is achieved.
The square root of √749 is approximately 27.370.
Square Root of 749 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 749 using the approximation method.
Step 1: We have to find the closest perfect square of √749.
The smallest perfect square less than 749 is 729 (27²) and the largest perfect square greater than 749 is 784 (28²). √749 falls somewhere between 27 and 28.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (749 - 729) ÷ (784 - 729) = 20 ÷ 55 ≈ 0.364.
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 + 0.364 ≈ 27.364, so the square root of 749 is approximately 27.364.
Common Mistakes and How to Avoid Them in the Square Root of 749
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods. 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 will primarily be considering the positive square root, as it is the required one.
For example: √50 ≈ 7.071, but there is also -7.071 which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root symbol 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 moving on to the next step.
For example: √(12+24) = √12 + √24 is incorrect. The right way is √(12+24) = √36 = 6.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to students is to simplify the numbers inside the square root. This is important so students do not make mistakes while identifying the approximate value.
For example: √50 ≈ 7 is incorrect; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly, as this will help them avoid mistakes.
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 either happen in subtracting or elsewhere.
Square Root of 749 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √749?
The area of the square is approximately 749 square units.
Explanation
The area of the square = side².
The side length is given as √749.
Area of the square = side² = √749 x √749 ≈ 27.370 × 27.370 ≈ 749.
Therefore, the area of the square box is approximately 749 square units.
Problem 2
A square-shaped building measuring 749 square feet is built; if each of the sides is √749, what will be the square feet of half of the building?
Approximately 374.5 square feet.
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 749 by 2 gives us approximately 374.5.
So, half of the building measures approximately 374.5 square feet.
Problem 3
Calculate √749 × 5.
Approximately 136.85.
Explanation
The first step is to find the square root of 749, which is approximately 27.370.
The second step is to multiply 27.370 by 5.
So, 27.370 × 5 ≈ 136.85.
Problem 4
What will be the square root of (729 + 20)?
The square root is approximately 27.037.
Explanation
To find the square root, we need to find the sum of (729 + 20). 729 + 20 = 749, and then √749 ≈ 27.370.
Therefore, the square root of (729 + 20) is approximately 27.370.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √749 units and the width ‘w’ is 30 units.
We find the perimeter of the rectangle as approximately 114.74 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√749 + 30) ≈ 2 × (27.370 + 30) ≈ 2 × 57.370 ≈ 114.74 units.
FAQ on Square Root of 749
1.What is √749 in its simplest form?
2.Mention the factors of 749.
3.Calculate the square of 749.
4.Is 749 a prime number?
5.749 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 749?
8.How do technology and digital tools in India support learning Algebra and Square Root of 749?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 749
- Square root: A square root is the inverse of squaring a number. For example, 4² = 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 expressed as a fraction, i.e., 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 more commonly used due to its applications in the real world. This is known as the principal square root.
- Decimal: If a number has a whole number and a fractional part, then it is called a decimal. For example: 27.37, 8.65, and 9.42 are decimals.
- Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 749 is 7 × 107.
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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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