Square Root of 3456

The square root is the inverse of the square of the number. 3456 is not a perfect square. The square root of 3456 is expressed in both radical and exponential form. In radical form, it is expressed as √3456, whereas in exponential form it is (3456)^(1/2). √3456 ≈ 58.78775, 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 3456 is broken down into its prime factors:

Step 1: Finding the prime factors of 3456. Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3: 2^6 × 3^4

Step 2: Now we have found the prime factors of 3456. The second step is to make pairs of those prime factors. Since 3456 is not a perfect square, the digits of the number can’t be grouped into pairs evenly. Therefore, calculating √3456 using prime factorization alone 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 3456, we need to group it as 56 and 34.

Step 2: Now we need to find n whose square is less than or equal to 34. The closest such number is 5, since 5 × 5 = 25. Now the quotient is 5, and after subtracting 25 from 34, the remainder is 9.

Step 3: Bring down 56, making the new dividend 956. Add the old divisor with the quotient 5 + 5 = 10, which will be our new divisor.

Step 4: The new divisor will be 10n. We need to find the value of n such that 10n × n ≤ 956. Considering n as 9, 10 × 9 × 9 = 810. Step 5: Subtract 810 from 956; the difference is 146, and the quotient so far is 59.

Step 6: Since the dividend is more significant than the divisor, add a decimal point and bring down two zeros, making the new dividend 14600.

Step 7: Find the new divisor, which is 118 because 1189 × 9 = 10701.

Step 8: Subtracting 10701 from 14600, we get the result 3899.

Step 9: Continue doing these steps until we reach the desired precision. So the square root of √3456 ≈ 58.78775

The approximation method is another way to find 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 3456 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3456. The smallest perfect square less than 3456 is 3364, and the largest perfect square greater than 3456 is 3481. √3456 falls somewhere between 58 and 59.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (3456 - 3364) / (3481 - 3364) = 92 / 117 ≈ 0.78632 Using the formula, we identified the decimal point of our square root. The next step is adding the integer part we found initially to the decimal number, which is 58 + 0.78632 ≈ 58.78775.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse is the square root, √16 = 4.

• Irrational number: An irrational number 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, but the principal square root is the positive one, which is commonly used.

• Prime factorization: Expressing a number as the product of its prime factors.

• Perfect square: A number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.