Square Root of 105.79

The square root is the inverse of the square of a number. 105.79 is not a perfect square. The square root of 105.79 is expressed in both radical and exponential forms.

In the radical form, it is expressed as √105.79, whereas (105.79)^(1/2) in the exponential form. √105.79 ≈ 10.2863, 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: -

1. Long division method 
2. 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 of 105.79 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 105.79, we need to group it as 05, 10, and .79.

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n is '3' because 3^2 = 9 is less than 10. Now the quotient is 3, after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 05, which is the new dividend. Add the old divisor with the same number, 3 + 3, to get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 105. Let us consider n as 1, now 6 × 1 × 1 = 6.

Step 6: Subtract 105 from 6, the difference is 99, and the quotient is 10

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down 79. Now the new dividend is 979.

Step 8: Now we need to find the new divisor that is 206 because 206 × 4 = 824.

Step 9: Subtracting 824 from 979, we get the result 155.

Step 10: Now the quotient is 10.2.

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

So the square root of √105.79 is approximately 10.29.

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 105.79 using the approximation method.

Step 1: Find the closest perfect squares to √105.79. The smallest perfect square less than 105.79 is 100, and the largest perfect square greater than 105.79 is 121. √105.79 falls between 10 and 11.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula: (105.79 - 100) ÷ (121 - 100) = 5.79 ÷ 21 ≈ 0.2757.

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 10 + 0.2757 ≈ 10.28,

so the square root of 105.79 is approximately 10.28.

• Square root: A square root is the inverse operation to squaring a number. For example, √16 = 4 because 4² = 16.

• Irrational number: An irrational number cannot be written as a simple fraction; its decimal form is non-repeating and non-terminating.

• Decimal: A decimal is a number that includes a decimal point followed by digits representing fractions of a base-ten number.

• Long division method: A technique used to find the square root of a non-perfect square by dividing and averaging.

• Approximation method: A method for estimating the square root of a number by finding the nearest perfect squares and making a calculated guess.