Square Root of 113

The square root is the inverse of the square of the number. 113 is not a perfect square. The square root of 113 is expressed in both radical and exponential form.

In the radical form, it is expressed as √113, whereas (113)^(1/2) in the exponential form. √113 ≈ 10.63014581273465, 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. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 113 is broken down into its prime factors:

Step 1: Finding the prime factors of 113

113 is a prime number, so it cannot be broken down further.

This means the prime factorization of 113 is simply 113 itself.

Since 113 is not a perfect square, calculating √113 using prime factorization directly is not feasible.

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 113, we need to group it as 13 and 1.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n as '1' because 1 × 1 is less than or equal to 1. Now the quotient is 1, and the remainder is 0 after subtracting 1 from 1.

Step 3: Bring down 13, which is the new dividend. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 13. Let us consider n as 6, now 26 × 6 = 156.

Step 6: Since 156 is greater than 13, we try n as 5, so 25 × 5 = 125, which is still greater. We try n as 4, so 24 × 4 = 96, which is less than 113.

Step 7: Subtract 96 from 113 to get the difference, which is 17. The quotient is 10.6.

Step 8: 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 1700.

Step 9: Continue this process until you reach the desired precision. The square root of √113 is approximately 10.63.

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 113 using the approximation method.

Step 1: Now we have to find the closest perfect square to √113. The smallest perfect square less than 113 is 100, and the largest perfect square greater than 113 is 121. √113 falls somewhere 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 (113 - 100) ÷ (121 - 100) ≈ 0.619. 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 + 0.63 ≈ 10.63, so the square root of 113 is approximately 10.63.

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

• Irrational number: An irrational number cannot be expressed as a simple fraction or ratio of two integers. It has a non-repeating, non-terminating decimal expansion.

• Principal square root: The principal square root is the non-negative square root of a number. Even though a number has both positive and negative square roots, the principal square root is the positive one.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 113 is an example of a prime number.

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point. For example, 10.63 is a decimal.