Square Root of 5.39

The square root is the inverse of the square of the number. 5.39 is not a perfect square. The square root of 5.39 is expressed in both radical and exponential form. In the radical form, it is expressed as √5.39, whereas (5.39)^(1/2) in the exponential form. √5.39 ≈ 2.321, 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:

• 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: To begin with, we need to group the numbers from right to left. In the case of 5.39, we need to group it as 5 and 39.

Step 2: Now we need to find n whose square is less than or equal to 5. We can say n is ‘2’ because 2 x 2 = 4 is less than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 39, making it the new dividend. Add the old divisor with the quotient 2 + 2 to 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 × n ≤ 139. Let us consider n as 3, now 4 x 3 x 3 = 36.

Step 6: Subtract 36 from 139, the difference is 103, and the quotient so far is 2.3.

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 zeroes to the dividend. Now the new dividend is 10300.

Step 8: Now we need to find the new divisor that is 23 because 463 x 23 = 10649.

Step 9: Subtracting 10649 from 10300, we get the result -349.

Step 10: Now the quotient is approximately 2.32.

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 5.39 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √5.39.

The perfect squares closest to 5.39 are 4 and 9. √5.39 falls somewhere between 2 and 3.

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

Going by the formula, (5.39 - 4) / (9 - 4) = 0.278.

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 2 + 0.278 = 2.278, so the square root of 5.39 is approximately 2.278.

• Square root: A square root is the inverse of a square. For 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.

• Decimal approximation: The approximate value of a non-perfect square number, often expressed in decimal form.

• Long division method: A step-by-step process used to find the square root of non-perfect square numbers.

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