Last updated on May 26th, 2025
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 675.
The square root is the inverse of the square of the number. 675 is not a perfect square. The square root of 675 is expressed in both radical and exponential forms. In the radical form, it is expressed as √675, whereas in exponential form, it is (675)^(1/2). √675 ≈ 25.98076, 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.
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the prime factorization method is not suitable. Instead, we use the long division method and approximation method. Let us now learn these methods:
The product of prime factors is the prime factorization of a number. Now let us look at how 675 is broken down into its prime factors.
Step 1: Finding the prime factors of 675 Breaking it down, we get 3 x 3 x 3 x 5 x 5: 3^3 x 5^2
Step 2: Now we found out the prime factors of 675. The second step is to make pairs of those prime factors. Since 675 is not a perfect square, the digits of the number can’t be grouped in a perfect pair.
Therefore, calculating 675 using prime factorization gives us √675 = 3 x 5 x √3 = 15√3.
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 675, we need to group it as 75 and 6.
Step 2: Now we need to find n whose square is closest to 6. We can say n is ‘2’ because 2 x 2 = 4, which is less than 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.
Step 3: Bring down the next pair, 75, to get the new dividend of 275. Add the previous divisor 2 with itself to get 4, which will be part of our new divisor.
Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n is less than or equal to 275. Let n be 6. Now, 46 x 6 = 276, which is too high. Try n = 5, we get 45 x 5 = 225.
Step 5: Subtract 225 from 275, the difference is 50, and the quotient so far is 25.
Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros to make it 5000.
Step 7: Now we need to find a new digit to append to 50 (the divisor) to make a number that multiplied by itself is close to 5000. If 509 x 9 = 4581, and that works, so we continue.
Step 8: Subtract 4581 from 5000 to get 419. Continue the long division to get more decimal places as needed.
So the square root of √675 is approximately 25.98.
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 675 using the approximation method.
Step 1: Now we have to find the closest perfect squares around √675. The smallest perfect square less than 675 is 625, and the largest perfect square greater than 675 is 676. √675 falls somewhere between 25 and 26.
Step 2: Apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (675 - 625) / (676 - 625) = 50 / 51 ≈ 0.98. Using this approximation, we add to the lower bound: 25 + 0.98 = 25.98.
So the square root of 675 is approximately 25.98.
Students make mistakes while finding the square root, like forgetting about the negative square root or skipping steps in the long division method. Let's look at some common mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √675?
The area of the square is approximately 675 square units.
The area of the square = side^2.
The side length is given as √675.
Area of the square = side^2 = √675 x √675 = 675
Therefore, the area of the square box is approximately 675 square units.
A square-shaped building measuring 675 square feet is built. If each of the sides is √675, what will be the square feet of half of the building?
337.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 675 by 2 = 337.5
So half of the building measures 337.5 square feet.
Calculate √675 x 5.
Approximately 129.9
The first step is to find the square root of 675, which is approximately 25.98.
The second step is to multiply 25.98 by 5.
So 25.98 x 5 ≈ 129.9.
What will be the square root of (650 + 25)?
The square root is approximately 26.
To find the square root, sum (650 + 25).
650 + 25 = 675, and then √675 ≈ 25.98.
Therefore, the square root of (650 + 25) is approximately ±25.98.
Find the perimeter of the rectangle if its length ‘l’ is √675 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 131.96 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√675 + 40)
Perimeter ≈ 2 × (25.98 + 40)
= 2 × 65.98
= 131.96 units.
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.
: He loves to play the quiz with kids through algebra to make kids love it.