Square Root of 706

The square root is the inverse of the square of the number. 706 is not a perfect square. The square root of 706 is expressed in both radical and exponential form. In the radical form, it is expressed as √706, whereas (706)^(1/2) in the exponential form. √706 ≈ 26.57066, 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, 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

The product of prime factors is the prime factorization of a number. Now let us look at how 706 is broken down into its prime factors.

Step 1: Finding the prime factors of 706

Breaking it down, we get 2 x 353: 2^1 x 353^1

Step 2: Now that we have found the prime factors of 706, the second step is to make pairs of those prime factors. Since 706 is not a perfect square, the digits of the number can't be grouped in pairs. Therefore, calculating 706 using prime factorization is not straightforward.

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 706, we need to group it as 06 and 7.

Step 2: Now we need to find n whose square is ≤ 7. We can say n is ‘2’ because 2 x 2 = 4 which is less than 7. Now the quotient is 2, after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 06, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 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 ≤ 306. Let us consider n as 7, now 47 x 7 = 329, which is too much, so we try n as 6.

Step 6: 46 x 6 = 276. Subtract 276 from 306, the difference is 30, and the quotient is 26.

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 zeros to the dividend. Now the new dividend is 3000.

Step 8: Now we need to find the new divisor that is 532 because 532 x 5 = 2660.

Step 9: Subtracting 2660 from 3000, we get the result 340.

Step 10: Now the quotient is 26.5.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √706 is approximately 26.57.

The approximation method is another method for finding the 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 706 using the approximation method.

Step 1: Now we have to find the closest perfect square of √706. The smallest perfect square less than 706 is 676 and the largest perfect square more than 706 is 729. √706 falls somewhere between 26 and 27.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (706 - 676) ÷ (729 - 676) = 30 ÷ 53 ≈ 0.566 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 26 + 0.57 = 26.57, so the square root of 706 is approximately 26.57.

• 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.

• 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 a principal square root.

• Long Division Method: A method used to find the square root of non-perfect square numbers by dividing and finding the quotient and remainder step by step.

• Approximation Method: A method used to estimate the square root by finding the closest perfect squares and determining a decimal value.