Square Root of 670

The square root is the inverse of the square of a number. 670 is not a perfect square. The square root of 670 is expressed in both radical and exponential form. In radical form, it is expressed as √670, whereas in exponential form, it is expressed as (670)^(1/2). √670 ≈ 25.88436, which is an irrational number because it cannot be expressed as a fraction 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 like 670, the long division method and approximation method are used. Let us now learn the following 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 670 is broken down into its prime factors:

Step 1: Finding the prime factors of 670 Breaking it down, we get 2 × 5 × 67: 2¹ × 5¹ × 67¹

Step 2: Now we found out the prime factors of 670. Since 670 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 670 using prime factorization directly is not feasible for finding its square root.

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 670, we need to group it as 70 and 6.

Step 2: Now we need to find n whose square is ≤ 6. We can say n is ‘2’ because 2 × 2 = 4 is less than or equal to 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.

Step 3: Now let us bring down 70, 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 4n. We need to find the value of n such that 4n × n is close to 270. Let n = 6, now 46 × 6 = 276

Step 5: Since 276 is more than 270, we try n = 5, so 45 × 5 = 225

Step 6: Subtracting 225 from 270, the difference is 45, and the quotient is 25.

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. The new dividend is 4500.

Step 8: Now we need to find n for 450n × n ≤ 4500. Let n = 8, so 508 × 8 = 4064

Step 9: Subtracting 4064 from 4500, we get the result 436.

Step 10: The quotient is now 25.8

Step 11: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √670 is approximately 25.88

The approximation method is another method for finding square roots. It's an easy method to find the square root of a given number. Now let us learn how to find the square root of 670 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √670. The smallest perfect square less than 670 is 625, and the largest perfect square greater than 670 is 676. √670 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (670 - 625) / (676 - 625) = 45 / 51 ≈ 0.882 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 25 + 0.882 = 25.882, so the square root of 670 is approximately 25.882.

• 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, it is usually the positive square root that is used due to its applications in the real world. This is known as the principal square root.
   
• Long division method: A method used to find the square root of non-perfect squares through a series of division steps.
   
• Approximation method: A method used to approximate the value of a square root by finding two perfect squares between which the number lies and then using a formula to find the decimal approximation.