Square Root of 946

The square root is the inverse of the square of the number. 946 is not a perfect square. The square root of 946 is expressed in both radical and exponential forms. In the radical form, it is expressed as √946, whereas (946)^(1/2) in the exponential form. √946 ≈ 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 946 is broken down into its prime factors.

Step 1: Finding the prime factors of 946 Breaking it down, we get 2 x 11 x 43: 2¹ x 11¹ x 43¹

Step 2: Now we found out the prime factors of 946. The second step is to make pairs of those prime factors. Since 946 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 946 using prime factorization for its square root isn't 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 946, we group it as 46 and 9.

Step 2: Now we need to find n whose square is less than or equal to 9. We can say n as ‘3’ because 3^2 = 9. Now the quotient is 3, and after subtracting 9 from 9, the remainder is 0.

Step 3: Now let us bring down 46, 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 ≤ 46. Let us consider n as 7, now 67 × 7 = 469.

Step 6: Subtract 469 from 460, the difference is -9, hence n should be lesser. Let's consider n as 6, now 66 × 6 = 396.

Step 7: Subtracting 396 from 460, the difference is 64, and the quotient is 36.

Step 8: 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 6400.

Step 9: Now we need to find the new divisor that is 367 because 367 × 7 = 2569.

Step 10: Subtracting 2569 from 6400 gives the result 3831.

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

So the square root of √946 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 946 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √946. The smallest perfect square less than 946 is 900, and the largest perfect square greater than 946 is 961. √946 falls somewhere between 30 and 31.

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

Applying the formula (946 - 900) ÷ (961 - 900) = 46 ÷ 61 ≈ 0.7541. 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 + 0.7541 ≈ 30.75, so the square root of 946 is approximately 30.75.

• 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.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is used in real-world applications, hence known as the principal square root.

• Prime factorization: Prime factorization is expressing a number as the product of its prime numbers.

• Decimal: A decimal is a number that consists of a whole number and a fractional part, separated by a decimal point, such as 7.86, 8.65, and 9.42.