Square Root of 1536

The square root is the inverse operation of squaring a number. 1536 is not a perfect square. The square root of 1536 is expressed in both radical and exponential form. In radical form, it is expressed as √1536, whereas in exponential form it is expressed as (1536)^(1/2). √1536 is an irrational number because it cannot be expressed as a fraction of two integers.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, 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 1536 is broken down into its prime factors.

Step 1: Finding the prime factors of 1536 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3: 2^8 x 3^1

Step 2: Now we found the prime factors of 1536. The second step is to make pairs of those prime factors. Since 1536 is not a perfect square, all the digits of the number can’t be grouped in pairs.

Therefore, calculating √1536 using prime factorization directly gives an approximate result.

The long division method is particularly used for non-perfect square numbers. 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 1536, we need to group it as 36 and 15.

Step 2: Now we need to find n whose square is less than or equal to 15. We can say n is ‘3’ because 3 x 3 = 9 which is less than 15. Now the quotient is 3, and after subtracting 9 from 15, the remainder is 6.

Step 3: Now let us bring down 36, which is the new dividend. Add the old divisor with itself (3 + 3) to get 6, which will be our new divisor.

Step 4: The new divisor is 60. We need to find a digit d such that 60d x d is less than or equal to 636.

Step 5: The suitable value of d is 1, as 601 x 1 = 601 is less than 636. Subtracting 601 from 636 gives a remainder of 35.

Step 6: 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 3500.

Step 7: Repeat the process to continue finding more decimal places as needed.

So the square root of √1536 is approximately 39.19.

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 1536 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1536.

The smallest perfect square less than 1536 is 1521 (39^2) and the largest perfect square more than 1536 is 1600 (40^2). √1536 falls somewhere between 39 and 40.

Step 2: Now, apply the approximation formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) (1536 - 1521) ÷ (1600 - 1521) = 15 / 79 = approximately 0.19

Using the formula, we identified the decimal point of our square root. The next step is adding 39 to the decimal number, which is 39 + 0.19 = 39.19.

So, the square root of 1536 is approximately 39.19.

• Square root: A square root is the inverse operation of squaring a number. 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 as a simple fraction, where both the numerator (p) and the denominator (q) are integers, and q is not zero.

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

• Prime factorization: Breaking down a number into its basic prime components. For example, the prime factorization of 1536 is 2^8 x 3.

• Decimal: A number that consists of a whole number and a fractional part separated by a decimal point, such as 39.19.