Square Root of 7.68

The square root is the inverse of the square of the number. 7.68 is not a perfect square. The square root of 7.68 is expressed in both radical and exponential forms. In the radical form, it is expressed as √7.68, whereas (7.68)^(1/2) in the exponential form. √7.68 ≈ 2.77, 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 prime factorization method is not applicable directly for non-perfect squares like 7.68. Instead, we use other methods like the long-division method and approximation method to find the square root of such numbers.

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 7.68, consider it as 7.68.

Step 2: Now, find a number whose square is less than or equal to 7. The closest is 2 since 2 x 2 = 4. The quotient is 2 and the remainder is 7 - 4 = 3.

Step 3: Bring down the decimal and the next digits (68) to make it 368. Double the quotient (2) to get the new divisor, which is 4.

Step 4: Find a number n such that 4n x n ≤ 368. The closest value is n = 7, as 47 x 7 = 329.

Step 5: Subtract 329 from 368 to get the remainder of 39.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros to make it 3900.

Step 7: Double the quotient (27) to get 54 as the new divisor. Find n such that 54n x n ≤ 3900.

Step 8: Continue the process until you achieve the desired level of accuracy.

So the square root of √7.68 ≈ 2.77.

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 7.68 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √7.68.

The smallest perfect square less than 7.68 is 4 and the largest perfect square greater than 7.68 is 9. √7.68 falls somewhere between 2 and 3.

Step 2: Now, we need to apply an approximation formula or estimate to find the value closer to the true square root of 7.68. Using a calculator or estimation: √7.68 ≈ 2.77.

• Square Root: A square root is the inverse operation of squaring a number. For example, the square root of 16 is 4, since 4^2 = 16.

• Irrational Number: A number that cannot be written as a simple fraction. An example is √7.68, which cannot be exactly expressed as a ratio of two integers.

• Principal Square Root: The non-negative square root of a number. For example, the principal square root of 9 is 3.

• Rational Number: A number that can be expressed as a fraction with integers in the numerator and denominator.

• Decimal: A number that includes a decimal point to represent a fraction of a base-10 number. For example, 7.68 is a decimal.