Square Root of 945

The square root is the inverse of the square of a number. 945 is not a perfect square. The square root of 945 is expressed in both radical and exponential form. In radical form, it is expressed as √945, whereas in exponential form, it is (945)^(1/2). √945 ≈ 30.74085, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 945 is broken down into its prime factors.

Step 1: Finding the prime factors of 945 Breaking it down, we get 3 x 3 x 3 x 5 x 7: 3^3 x 5 x 7

Step 2: Now we have found the prime factors of 945. The second step is to make pairs of those prime factors. Since 945 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √945 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 945, we need to group it as 45 and 9.

Step 2: Now we need to find n whose square is close to 9. We can say n is ‘3’ because 3 x 3 is 9. Now the quotient is 3, and after subtracting 9 - 9, the remainder is 0.

Step 3: Now bring down 45, which is the new dividend. Add the old divisor with the same number 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 45. Let us consider n as 7, now 6 x 7 x 7 = 294.

Step 6: Subtract 45 from 294; since 294 is more, reduce n to 6. The difference is 9, and the quotient is 36.

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 900.

Step 8: Now we need to find the new divisor that is 367 because 367 x 2 = 734.

Step 9: Subtracting 734 from 900, we get the result 166.

Step 10: Now the quotient is 30.7.

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

So the square root of √945 is approximately 30.74.

The 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 945 using the approximation method.

Step 1: Now we have to find the closest perfect square of √945. The smallest perfect square less than 945 is 900, and the largest perfect square greater than 945 is 961. √945 falls somewhere between 30 and 31.

Step 2: Now we need to apply the formula that is (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square).

Using the formula (945 - 900) / (961 - 900) = 0.738. 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 30 + 0.738 = 30.738.

So the square root of 945 is approximately 30.74.

• Square root: A square root is the inverse of a square. Example: 4² = 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.

• Radical form: A number is expressed with a square root symbol, such as √x.

• Approximation method: A technique used to find the approximate square root of a non-perfect square number.

• Prime factorization: Breaking down a composite number into its prime factors.