Square Root of 756

The square root is the inverse of the square of the number. 756 is not a perfect square. The square root of 756 is expressed in both radical and exponential form. In the radical form, it is expressed as √756, whereas (756)^(1/2) is in the exponential form. √756 ≈ 27.495, 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 756 is broken down into its prime factors.

Step 1: Finding the prime factors of 756 Breaking it down, we get 2 x 2 x 3 x 3 x 3 x 7: 2^2 x 3^3 x 7

Step 2: Now we found out the prime factors of 756. The second step is to make pairs of those prime factors. Since 756 is not a perfect square, the digits of the number can’t be grouped in pair.

Therefore, calculating 756 using prime factorization is impossible.

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 756, we need to group it as 56 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n is ‘2’ because 2 × 2 = 4. The quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 56, making it the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. Now we need to find n such that 4n × n is less than or equal to 356. Let us consider n as 7, so 47 × 7 = 329.

Step 5: Subtract 329 from 356 to find the difference, which is 27, and the quotient becomes 27.

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 2700.

Step 7: Now we need to find the new divisor. We estimate n as 5, because 545 × 5 = 2725.

Step 8: Subtracting 2725 from 2700 results in a remainder of -25.

Step 9: Continue refining n and divisor until you achieve the desired precision. The quotient approximates to 27.49.

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 756 using the approximation method.

Step 1: Now we have to find the closest perfect square of √756.

The smallest perfect square less than 756 is 729, and the largest perfect square greater than 756 is 784. √756 falls somewhere between 27 and 28.

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

Using the formula (756 - 729) ÷ (784 - 729) = 27 ÷ 55 ≈ 0.49 Using this approximation, we identified the decimal part of our square root. The next step is adding the value we got initially to the decimal number, which gives 27 + 0.49 = 27.49.

So the square root of 756 is approximately 27.49.

• 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, but the positive square root is more commonly used, known as the principal square root.

• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 756 is 2² × 3³ × 7.

• Long division method: A technique used to find the square root of non-perfect squares by dividing the number step-by-step to reach an approximate value.