190 LearnersLast updated on May 26th, 2025
Square Root of 74
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 74.
What is the Square Root of 74?
The square root is the inverse of the square of the number. 74 is not a perfect square. The square root of 74 is expressed in both radical and exponential form.
In the radical form, it is expressed as √74, whereas (74)(1/2) in the exponential form. √74 ≈ 8.60233, 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 74
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, methods like 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 74 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 74 is broken down into its prime factors.
Step 1: Finding the prime factors of 74 Breaking it down, we get 2 x 37: 21 x 371
Step 2: Now we found out the prime factors of 74. The second step is to make pairs of those prime factors. Since 74 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 74 using prime factorization is not straightforward.
Square Root of 74 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 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 74, we need to group it as 74.
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, after subtracting 4 from 7, the remainder is 3.
Step 3: Bring down 4, making the new dividend 34. Double the divisor, 2, to get 4.
Step 4: The next step is finding 4n × n ≤ 34. Let us consider n as 8; now 48 x 8 = 384, which is more than 34, so we try n as 7, and 47 x 7 = 329, which works.
Step 5: Subtract 329 from 340, the difference is 11, and the quotient is 8.6
Step 6: 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 remainder. Now the new dividend is 1100.
Step 7: Find the new divisor, which is 86, because 860 x 1 = 860
Continue these steps until you get the desired decimal places. So the square root of √74 ≈ 8.602
Square Root of 74 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 74 using the approximation method.
Step 1: Now we have to find the closest perfect square of √74. The smallest perfect square less than 74 is 64 and the largest perfect square greater than 74 is 81. √74 falls somewhere between 8 and 9.
Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square).
Using the formula (74 - 64) ÷ (81 - 64) = 0.588
Adding the approximate value to the lower boundary of 8, we get 8 + 0.588 ≈ 8.588, so the square root of 74 is approximately 8.588.
Common Mistakes and How to Avoid Them in the Square Root of 74
Students make mistakes while finding the square root, like forgetting about the negative square root or skipping steps in the long division methods. Let us look at a few common mistakes.
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 usually consider only the positive square root, as it is the practical one.
For example: √50 = 7.07, 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 students make. To resolve this, students need to simplify the numbers inside the square root first.
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 is to simplify the numbers inside the square root. This prevents 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 understand the difference between cube root and square root, as this helps 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, students might skip a step accidentally, resulting in a wrong answer. This may happen during subtraction or elsewhere in the process.
Square Root of 74 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √90?
The area of the square is approximately 90 square units.
Explanation
The area of the square = side2.
The side length is given as √90.
Area of the square = side2 = √90 x √90 = 9.486 x 9.486 ≈ 90.
Therefore, the area of the square box is approximately 90 square units.
Problem 2
A square-shaped building measuring 74 square feet is built; if each of the sides is √74, what will be the square feet of half of the building?
37 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 74 by 2 = we get 37
So half of the building measures 37 square feet.
Problem 3
Calculate √74 x 5.
Approximately 43.01
Explanation
The first step is to find the square root of 74, which is approximately 8.602.
The second step is to multiply 8.602 by 5.
So 8.602 x 5 ≈ 43.01
Problem 4
What will be the square root of (64 + 10)?
The square root is 9.
Explanation
To find the square root, we need to find the sum of (64 + 10). 64 + 10 = 74, and then √74 ≈ 8.602.
Therefore, the square root of (64 + 10) is approximately ±8.602.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √74 units and the width ‘w’ is 10 units.
We find the perimeter of the rectangle as approximately 37.204 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√74 + 10) ≈ 2 × (8.602 + 10) ≈ 2 × 18.602 ≈ 37.204 units.
FAQ on Square Root of 74
1.What is √74 in its simplest form?
2.Mention the factors of 74.
3.Calculate the square of 74.
4.Is 74 a prime number?
5.74 is divisible by?
Important Glossaries for the Square Root of 74
- Square root: A square root is the inverse of a square. Example: 42 = 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.
- Long division method: A method used to find the square root of non-perfect squares by dividing numbers into groups of two digits.
- Perfect square: A number that is the square of an integer. For example, 64 is a perfect square because it equals 82.
- Approximation method: A method used to estimate the square root of a number by finding the closest perfect squares and interpolating between them.
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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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
