127 LearnersLast updated on May 26th, 2025
Square Root of 785
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 various fields like vehicle design, finance, etc. Here, we will discuss the square root of 785.
What is the Square Root of 785?
The square root is the inverse of the square of a number. 785 is not a perfect square. The square root of 785 is expressed in both radical and exponential form. In radical form, it is expressed as √785, whereas in exponential form it is (785)^(1/2). √785 ≈ 28.0179, 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 785
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, 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 785 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 785 is broken down into its prime factors.
Step 1: Finding the prime factors of 785 Breaking it down, we get 5 x 157: 5^1 x 157^1
Step 2: Now we found out the prime factors of 785. The second step is to make pairs of those prime factors. Since 785 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 785 using prime factorization directly is not feasible.
Square Root of 785 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 785, we group it as 85 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, which is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.
Step 3: Now bring down 85, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.
Step 4: The new divisor becomes 20n, and we need to find the value of n.
Step 5: The next step is finding 40n ≤ 385. Let us consider n as 7, now 40 x 7 = 280.
Step 6: Subtract 280 from 385, the difference is 105, and the quotient becomes 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 10500.
Step 8: Now, consider a new divisor, which is 40n, and find n such that 407n ≤ 10500.
Step 9: By trial, 407 x 7 = 2849, which fits.
Step 10: Subtracting 2849 from 10500, we get the remainder 7631.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue until the remainder is zero.
So the square root of √785 ≈ 28.02
Square Root of 785 by Approximation Method
The approximation method is an easy method for finding square roots. Let's learn how to find the square root of 785 using this method.
Step 1: Find the closest perfect squares to √785.
The smallest perfect square less than 785 is 729, and the largest perfect square greater than 785 is 841. √785 falls somewhere between 27 and 29.
Step 2: Now apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Applying the formula: (785 - 729) / (841 - 729) = 56 / 112 = 0.5
Using this, we estimate the decimal part of our square root. The next step is adding the value we got initially to the decimal number: 27 + 0.5 = 27.5, so the approximate square root of 785 is 28.02
Common Mistakes and How to Avoid Them in the Square Root of 785
Students make mistakes while finding square roots, such as forgetting about the negative square root or skipping long division steps. Let us look at a few common mistakes in detail.
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 for practical 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
Misplacement of the square root symbol is a common error. Simplifying the numbers inside the square root before proceeding ensures accuracy.
For example: √(12+24) = √36 is correct, whereas √12 + √24 is incorrect.
Mistake 3
Not finding the correct value of a non-perfect square
Students should simplify numbers inside the square root correctly to avoid errors in finding approximate values.
For example: √50 = 7 is wrong; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
It is crucial to understand the difference between cube root and square root symbols to avoid confusion.
For example: √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
The long division method involves many steps, and skipping any can result in incorrect answers. This might occur during subtraction or elsewhere in the process.
Square Root of 785 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √785?
The area of the square is approximately 785 square units.
Explanation
The area of the square = side^2.
The side length is given as √785.
Area of the square = (√785) x (√785) ≈ 28.0179 x 28.0179 ≈ 785
Therefore, the area of the square box is approximately 785 square units.
Problem 2
A square-shaped garden measuring 785 square feet is built; if each of the sides is √785, what will be the square feet of half of the garden?
392.5 square feet
Explanation
Since the garden is square-shaped, we can divide the given area by 2 to find the area of half of the garden.
Dividing 785 by 2 = 392.5
So, half of the garden measures 392.5 square feet.
Problem 3
Calculate √785 x 5.
Approximately 140.09
Explanation
The first step is to find the square root of 785, which is approximately 28.02.
Multiply 28.02 by 5.
So, 28.02 x 5 ≈ 140.09
Problem 4
What will be the square root of (784 + 1)?
The square root is 28.02
Explanation
To find the square root, we calculate the sum of (784 + 1). 784 + 1 = 785, and then √785 ≈ 28.02.
Therefore, the square root of (784 + 1) is approximately 28.02.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √785 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 132.04 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√785 + 38) ≈ 2 × (28.02 + 38) ≈ 2 × 66.02 ≈ 132.04 units.
FAQ on Square Root of 785
1.What is √785 in its simplest form?
2.Mention the factors of 785.
3.Calculate the square of 785.
4.Is 785 a prime number?
5.785 is divisible by?
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 785?
8.How do technology and digital tools in Oman support learning Algebra and Square Root of 785?
9.Does learning Algebra support future career opportunities for students in Oman?
Important Glossaries for the Square Root of 785
- 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 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.
- Prime factorization: Breaking down a composite number into its prime factors. For example, the prime factorization of 785 is 5 x 157.
- Long division method: A technique used to find roots, especially for numbers that are not perfect squares, involving division and subtraction steps.
- Approximation method: A method to estimate the value of a square root by using nearby perfect squares and refining the estimate using a formula.
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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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