Square Root of 74

The square root is the inverse of the square of the number. 74 is not a perfect square. The square root of 74 is expressed in both radical and exponential form.

In the radical form, it is expressed as √74, whereas (74)^(1/2) in the exponential form. √74 ≈ 8.60233, 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, for non-perfect square numbers, methods like the long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 74 is broken down into its prime factors.

Step 1: Finding the prime factors of 74 Breaking it down, we get 2 x 37: 2¹ x 37¹

Step 2: Now we found out the prime factors of 74. The second step is to make pairs of those prime factors. Since 74 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 74 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we 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 74, we need to group it as 74.

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 x 2 = 4, which is less than 7. Now the quotient is 2, after subtracting 4 from 7, the remainder is 3.

Step 3: Bring down 4, making the new dividend 34. Double the divisor, 2, to get 4.

Step 4: The next step is finding 4n × n ≤ 34. Let us consider n as 8; now 48 x 8 = 384, which is more than 34, so we try n as 7, and 47 x 7 = 329, which works.

Step 5: Subtract 329 from 340, the difference is 11, and the quotient is 8.6

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 zeros to the remainder. Now the new dividend is 1100.

Step 7: Find the new divisor, which is 86, because 860 x 1 = 860

Continue these steps until you get the desired decimal places. So the square root of √74 ≈ 8.602

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 74 using the approximation method.

Step 1: Now we have to find the closest perfect square of √74. The smallest perfect square less than 74 is 64 and the largest perfect square greater than 74 is 81. √74 falls somewhere between 8 and 9.

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

Using the formula (74 - 64) ÷ (81 - 64) = 0.588

Adding the approximate value to the lower boundary of 8, we get 8 + 0.588 ≈ 8.588, so the square root of 74 is approximately 8.588.

• 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.

• Long division method: A method used to find the square root of non-perfect squares by dividing numbers into groups of two digits.

• Perfect square: A number that is the square of an integer. For example, 64 is a perfect square because it equals 8².

• Approximation method: A method used to estimate the square root of a number by finding the closest perfect squares and interpolating between them.