Square Root of 103

The square root is the inverse of the square of the number. 103 is not a perfect square. The square root of 103 is expressed in both radical and exponential form.

In radical form, it is expressed as √103, whereas (103)^1/2 in exponential form. √103 ≈ 10.14889, 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 like 103, methods such as 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. This method involves finding the closest perfect square number for the given number. Let us now learn how to find the square root of 103 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 103, we treat it as a single group because it has only three digits.

Step 2: Now we need to find n whose square is closest to 10. We can say n is ‘3’ because 3 × 3 = 9, which is less than 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Bring down 3 next to the remainder, making it 13. Add the old divisor with the same number, 3 + 3, to get 6 as the new divisor.

Step 4: The new divisor is 6n. We need to find the value of n. Try n = 2, so 6 × 2 = 12, which is less than or equal to 13.

Step 5: Subtract 12 from 13, and the remainder is 1. The quotient becomes 10.2.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point and bring down two zeros, making the new dividend 100.

Step 7: The new divisor will be 62. Try n = 1, so 62 × 1 = 62.

Step 8: Subtract 62 from 100, leaving a remainder of 38.

Step 9: Continue doing these steps until we get the desired precision.

We find the square root of √103 ≈ 10.14889.

The approximation method is another approach 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 103 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 103. The smallest perfect square less than 103 is 100, and the largest perfect square greater than 103 is 121. √103 falls somewhere between 10 and 11.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (103 - 100) / (121 - 100) = 3 / 21 ≈ 0.143

Adding this value to the square root of the smaller perfect square, we get 10 + 0.143 = 10.143.

This approximation shows √103 ≈ 10.143.

• 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, which 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

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4².

• Approximation: Approximation is the process of finding a value that is close enough to the correct answer, usually within a specified tolerance.