Square Root of 466

The square root is the inverse of the square of the number. 466 is not a perfect square. The square root of 466 is expressed in both radical and exponential form. In the radical form, it is expressed as √466, whereas in the exponential form it is written as (466)^(1/2). √466 ≈ 21.58703, 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 typically used for non-perfect square numbers; instead, 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 466 is broken down into its prime factors.

Step 1: Finding the prime factors of 466 Breaking it down, we get 2 × 233, where 233 is a prime number.

Step 2: Now we have found out the prime factors of 466. Since 466 is not a perfect square, its digits cannot be grouped into pairs.

Therefore, calculating 466 using prime factorization to find an exact square root is not possible.

The long division method is particularly useful 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 466, we group it as 66 and 4.

Step 2: Now we need to find n whose square is 4. We can say n is '2' because 2 × 2 is equal to 4. Now the quotient is 2, and after subtracting 4-4, the remainder is 0.

Step 3: Let us bring down 66, which is the new dividend. Add the old divisor with the same number: 2 + 2 gives us 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, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 66. Let us consider n as 1, since 41 × 1 = 41.

Step 6: Subtract 66 from 41; the difference is 25. The quotient is 21.

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 2500.

Step 8: Now we need to find the new divisor that is 5 because 425 × 5 = 2125.

Step 9: Subtracting 2125 from 2500, we get the result 375.

Step 10: Now the quotient is 21.5.

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 √466 is approximately 21.58.

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 466 using the approximation method.

Step 1: We have to find the closest perfect squares around √466. The smallest perfect square less than 466 is 441 (because 21² = 441), and the largest perfect square greater than 466 is 484 (because 22² = 484). √466 falls between 21 and 22.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (466 - 441) / (484 - 441) = 25 / 43 ≈ 0.58.

Using the formula, we identified the decimal point of our square root. The next step is adding the whole number we got initially to the decimal number: 21 + 0.58 = 21.58.

So the square root of 466 is approximately 21.58.

• Square root: A square root is the inverse of a square. For 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 typically used due to its applications in the real world. That is why it is known as the principal square root.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Factorization: Factorization is the process of breaking down a number into its prime factors. For example, the prime factorization of 466 is 2 × 233.