130 LearnersLast updated on May 26th, 2025
Square Root of 777
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 fields like vehicle design, finance, etc. Here, we will discuss the square root of 777.
What is the Square Root of 777?
The square root is the inverse of the square of the number. 777 is not a perfect square. The square root of 777 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √777, whereas (777)^(1/2) is the exponential form. √777 ≈ 27.8747, 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 777
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 777 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 777 is broken down into its prime factors:
Step 1: Finding the prime factors of 777. Breaking it down, we get 3 x 7 x 37: 3¹ x 7¹ x 37¹.
Step 2: Now we have found the prime factors of 777. Since 777 is not a perfect square, the digits of the number can’t be grouped into pairs.
Therefore, calculating the square root of 777 using prime factorization is not feasible.
Square Root of 777 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 777, we need to group it as 77 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 is less than or equal to 7. Now the quotient is 2; after subtracting 4 from 7, the remainder is 3.
Step 3: Now let us bring down 77, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.
Step 4: The new divisor will be 40 (as 4 becomes 40 by adding a digit). We need to find the value of n such that 40n x n ≤ 377.
Step 5: Let's try n = 9. Now 409 x 9 = 3681.
Step 6: Subtract 3681 from 3770, the difference is 89, and the quotient is 27.9.
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 8900.
Step 8: Now we need to find the new divisor, which is 558 because 558 x 8 = 4464.
Step 9: Subtracting 4464 from 8900, we get the result 4436.
Step 10: Now the quotient is 27.87.
Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.
So the square root of √777 is approximately 27.8747.
Square Root of 777 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 777 using the approximation method.
Step 1: Now we have to find the closest perfect square of √777.
The smallest perfect square less than 777 is 729 (27²), and the largest perfect square greater than 777 is 784 (28²). √777 falls somewhere between 27 and 28.
Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square) Using the formula: (777 - 729) / (784 - 729) = 48 / 55 ≈ 0.8727
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 + 0.8747 ≈ 27.8747, so the square root of 777 is approximately 27.8747.
Common Mistakes and How to Avoid Them in the Square Root of 777
Students often make mistakes while finding the square root, such as forgetting about the negative square root, skipping steps in the long division method, etc. Now let us look at a few of these mistakes 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 usually consider only the positive square root, as it is the required one.
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 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 to the next step.
For example: √(12 + 24) = √36 ≠ √12 + √24.
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 that students do not make mistakes while identifying the approximate value.
For example: √50 ≈ 7.071, not 7.
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, which results in a wrong answer at the end of the process. This may happen during subtraction or elsewhere.
Square Root of 777 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √777?
The area of the square is approximately 777 square units.
Explanation
The area of the square = side².
The side length is given as √777.
Area of the square = side² = √777 x √777 ≈ 777.
Therefore, the area of the square box is approximately 777 square units.
Problem 2
A square-shaped building measuring 777 square feet is built; if each of the sides is √777, what will be the square feet of half of the building?
388.5 square feet
Explanation
Since the building is square-shaped, we can simply divide the area by 2.
Dividing 777 by 2 gives us 388.5.
So half of the building measures 388.5 square feet.
Problem 3
Calculate √777 x 5.
Approximately 139.3735
Explanation
The first step is to find the square root of 777, which is approximately 27.8747.
The second step is to multiply 27.8747 by 5.
So 27.8747 x 5 ≈ 139.3735.
Problem 4
What will be the square root of (777 + 7)?
The square root is approximately 28.2843.
Explanation
To find the square root, we need to find the sum of (777 + 7), which equals 784.
The square root of 784 is 28. Therefore, the square root of (777 + 7) is ±28, but we usually refer to the positive value.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √777 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 131.7494 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√777 + 38) ≈ 2 × (27.8747 + 38) ≈ 2 × 65.8747 ≈ 131.7494 units.
FAQ on Square Root of 777
1.What is √777 in its simplest form?
2.Mention the factors of 777.
3.Calculate the square of 777.
4.Is 777 a prime number?
5.Is 777 divisible by 3?
6.How does learning Algebra help students in Oman make better decisions in daily life?
7.How can cultural or local activities in Oman support learning Algebra topics such as Square Root of 777?
8.How do technology and digital tools in Oman support learning Algebra and Square Root of 777?
9.Does learning Algebra support future career opportunities for students in Oman?
Important Glossaries for the Square Root of 777
- Square root: A square root is the inverse of a square. 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 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 more commonly used due to its practicality in real-world applications. This is why it is known as the principal square root.
- Prime factorization: The expression of a number as a product of its prime factors. For example, the prime factorization of 777 is 3 x 7 x 37.
- Long division method: A method used to divide two numbers, often used for finding square roots of non-perfect square 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.
Fun Fact
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