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:

• Prime factorization method
• Long division method
• Approximation method

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.

• 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 cannot be written in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Prime factorization: The process of expressing a number as a product of its prime factors. Example: 675 = 3^3 x 5^2.
   
• Approximation: A method used to find an approximate value of a mathematical expression. It is used when exact values are difficult to determine.
   
• Decimal: A number that includes a whole number and a fractional part separated by a decimal point. Example: 7.86, 8.65, and 9.42 are decimals.