118 LearnersLast updated on May 26th, 2025
Square Root of 782
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 782.
What is the Square Root of 782?
The square root is the inverse of the square of the number. 782 is not a perfect square. The square root of 782 is expressed in both radical and exponential form. In the radical form, it is expressed as √782, whereas (782)^(1/2) in the exponential form. √782 = 27.96426, 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 782
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 782 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 782 is broken down into its prime factors.
Step 1: Finding the prime factors of 782 Breaking it down, we get 2 x 17 x 23.
Step 2: Now we found out the prime factors of 782. The second step is to make pairs of those prime factors. Since 782 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating the square root of 782 using prime factorization is not feasible.
Square Root of 782 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 782, we need to group it as 82 and 7.
Step 2: Now we need to find n whose square is less than or equal to 7. We can say n as ‘2’ because 2 x 2 = 4 is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.
Step 3: Now let us bring down 82, 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 the sum of the dividend and quotient. Now we get 4n as the new divisor, we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 382. Let us consider n as 9, now 49 x 9 = 441.
Step 6: Subtracting gives us a negative remainder, so try n as 8, 48 x 8 = 384.
Step 7: Subtract 382 from 384, and the difference is negative, indicating 8 is still too high, so try 47 x 8 = 376, giving a remainder of 6.
Step 8: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 600.
Step 9: Find the new divisor. Since 479 x 9 = 4311 is too high, try 478 x 8 = 3824.
Step 10: Subtracting gives a remainder of 176. Continue with these steps.
So the square root of √782 is approximately 27.96.
Square Root of 782 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 782 using the approximation method.
Step 1: Now we have to find the closest perfect square of √782.
The smallest perfect square less than 782 is 729, and the largest perfect square greater than 782 is 841. √782 falls somewhere between 27 and 29.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (782 - 729) / (841 - 729) = 53 / 112 ≈ 0.4732.
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 + 0.4732 ≈ 27.97.
So, the square root of 782 is approximately 27.97.
Common Mistakes and How to Avoid Them in the Square Root of 782
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division steps. 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 does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the commonly required 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 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 an incorrect method. The correct way 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 wrong, 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 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 an incorrect answer at the end. This may happen during subtraction or elsewhere.
Square Root of 782 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √782?
The area of the square is 782 square units.
Explanation
The area of the square = side^2.
The side length is given as √782.
Area of the square = side^2 = √782 x √782 = 782.
Therefore, the area of the square box is 782 square units.
Problem 2
A square-shaped building measuring 782 square feet is built; if each of the sides is √782, what will be the square feet of half of the building?
391 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 782 by 2 = we get 391.
So half of the building measures 391 square feet.
Problem 3
Calculate √782 x 5.
139.82
Explanation
The first step is to find the square root of 782, which is approximately 27.96.
The second step is to multiply 27.96 by 5.
So 27.96 x 5 = 139.82.
Problem 4
What will be the square root of (782 + 18)?
The square root is 28
Explanation
To find the square root, we need to find the sum of (782 + 18). 782 + 18 = 800, and then √800 ≈ 28.28.
Therefore, the square root of (782 + 18) is approximately ±28.28.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √782 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle to be 131.92 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√782 + 38) = 2 × (27.96 + 38) = 2 × 65.96 = 131.92 units.
FAQ on Square Root of 782
1.What is √782 in its simplest form?
2.Mention the factors of 782.
3.Calculate the square of 782.
4.Is 782 a prime number?
5.782 is divisible by?
6.How does learning Algebra help students in Qatar make better decisions in daily life?
7.How can cultural or local activities in Qatar support learning Algebra topics such as Square Root of 782?
8.How do technology and digital tools in Qatar support learning Algebra and Square Root of 782?
9.Does learning Algebra support future career opportunities for students in Qatar?
Important Glossaries for the Square Root of 782
- Square root: A square root is the inverse of a square. 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
- Prime factorization: The process of breaking down a number into its smallest prime factors. For example, the prime factorization of 782 is 2 x 17 x 23.
- Long division method: A method used for finding the square root of non-perfect squares by grouping numbers and using division to approximate the square root.
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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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