Square Root of 745

The square root is the inverse of the square of the number. 745 is not a perfect square. The square root of 745 is expressed in both radical and exponential form. In the radical form, it is expressed as √745, whereas 745^(1/2) in the exponential form. √745 ≈ 27.288, 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 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 745 is broken down into its prime factors.

Step 1: Finding the prime factors of 745 Breaking it down, we get 5 x 149: 5¹ x 149¹

Step 2: Now we found out the prime factors of 745. Since 745 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 745 using prime factorization is not feasible for finding its square root.

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 745, we need to group it as 45 and 7.

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

Step 3: Now let us bring down 45, making the new dividend 345. Add the old divisor with the same number, 2 + 2, we get 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, we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 345. Let us consider n as 7, now 47 x 7 = 329.

Step 6: Subtract 329 from 345; the difference is 16, and the quotient is 27.

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 zeros to the dividend. Now the new dividend is 1600.

Step 8: Now we need to find the new divisor that is 547 because 547 x 2 = 1094.

Step 9: Subtracting 1094 from 1600, we get the result 506.

Step 10: Now the quotient becomes 27.2.

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

So the square root of √745 ≈ 27.288.

The approximation method is another method for finding the 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 745 using the approximation method.

Step 1: We need to find the closest perfect squares around 745.

The smallest perfect square less than 745 is 729, and the largest perfect square greater than 745 is 784. √745 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, (745 - 729) / (784 - 729) = 16 / 55 ≈ 0.291. 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 27 + 0.291 ≈ 27.291.

So the square root of 745 is approximately 27.291.

• Square root: A square root is the inverse of a square. 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 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, the positive square root is more prominent due to its uses in the real world. This is known as the principal square root.

• Integer: Integers are a combination of whole numbers and their negatives, for example, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, etc.

• Decimal: A decimal represents a number with a whole part and a fractional part, for example, 7.86, 8.65, and 9.42 are decimals.