Square Root of 7/3

The square root is the inverse of the square of a number. 7/3 is not a perfect square. The square root of 7/3 is expressed in both radical and exponential form. In the radical form, it is expressed as √(7/3), whereas (7/3)^(1/2) in the exponential form. √(7/3) approximately equals 1.5275, 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:

• Long division method
• Approximation method

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: Convert the fraction 7/3 into a decimal, approximately 2.333.

Step 2: To begin with, we need to group the numbers from right to left. In the case of 2.333, we need to group it as 2.33 and 0.03.

Step 3: Now we need to find n whose square is less than or equal to 2.33. We can choose n as ‘1’ because 1 × 1 is lesser than or equal to 2.33. Now the quotient is 1 and the remainder is 1.33.

Step 4: Bring down 0.03 to make it 133. Add the old divisor with the same number 1 + 1 to get 2 which will be our new divisor.

Step 5: The next step is finding 2n × n ≤ 133. Let us consider n as 6, now 26 × 6 = 156 which is more than 133, so we choose n as 5.

Step 6: Subtract 133 - 125 (25 × 5) to get 8 and the quotient is 1.5.

Step 7: Since the dividend is less than the divisor, we add a decimal point. Adding a decimal point allows us to add two zeroes to the dividend. Now the new dividend is 800.

Step 8: Find the new divisor, which is 31, because 31 × 2 gives a number less than 80.

Step 9: Subtract 800 - 775 (31 × 25) to get the remainder 25.

Step 10: Continue these steps until you get two decimal places.

The square root of √(7/3) is approximately 1.5275.

The approximation method is another method for finding square roots, and 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 7/3 using the approximation method.

Step 1: Convert 7/3 into a decimal, approximately 2.333.

Step 2: Find the closest perfect squares around 2.333. The smallest perfect square is 1 and the largest is 4. √(7/3) falls somewhere between 1 and 2.

Step 3: Apply the interpolation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).Going by the formula: (2.333 - 1) / (4 - 1) = 0.444.

Step 4: Using the formula, we identify the decimal point of our square root. Adding this to the smallest integer, 1 + 0.444 = 1.444, so the square root of 7/3 is approximately 1.5275.

• 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; however, it is the positive square root that is more commonly used due to its applications in the real world. This is known as the principal square root.

• Rational number: A number that can be expressed as the quotient or fraction p/q of two integers, where the denominator q is not zero.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.