Square Root of 991

The square root is the inverse of the square of the number. 991 is not a perfect square. The square root of 991 is expressed in both radical and exponential form. In the radical form, it is expressed as √991, whereas \(991^{1/2}\) in the exponential form. √991 ≈ 31.496, 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 991, 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 product of prime factors is the prime factorization of a number. Now let us look at how 991 is broken down into its prime factors. Step 1: Finding the prime factors of 991 991 is a prime number and cannot be broken down into smaller factors. Therefore, calculating √991 using prime factorization is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we find the square root step by step: Step 1: To begin with, we need to group the numbers from right to left. In the case of 991, we need to group it as 91 and 9. Step 2: Now we need to find n whose square is less than or equal to 9. We can choose n as '3' because \(3 \times 3 = 9\), which is equal to 9. Now the quotient is 3 after subtracting 9 from 9, leaving the remainder as 0. Step 3: Now let us bring down 91 which is the new dividend. Add the old divisor with the same number 3 + 3, we get 6 which will be our new divisor. Step 4: The new divisor will be 6n. We need to find the value of n such that \(6n \times n \leq 91\). Let us consider n as 1, now \(61 \times 1 = 61\). Step 5: Subtract 91 from 61, the difference is 30, and the quotient becomes 31. Step 6: 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 3000. Step 7: Now we need to find the new divisor which is 631, because \(631 \times 4 = 2524\). Step 8: Subtracting 2524 from 3000, we get the result 476. Step 9: The quotient is 31.4. Continue these steps until we get the desired decimal precision. So the square root of √991 ≈ 31.496.

The approximation method is another method for finding square roots, and 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 991 using the approximation method. Step 1: Now we have to find the closest perfect squares to √991. The smallest perfect square less than 991 is 961 (31^2) and the largest perfect square greater than 991 is 1024 (32^2). √991 falls somewhere between 31 and 32. Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (991 - 961) / (1024 - 961) = 30/63 ≈ 0.476 Using the formula, we identified that the decimal point of our square root is approximately 0.476. The next step is adding the value we got initially to the decimal number, which is 31 + 0.476 ≈ 31.476. Therefore, the square root of 991 is approximately 31.476.

Square Root: A square root is the inverse of a square. For example, \(4^2 = 16\) and the inverse of the square is the square root, so \(√16 = 4\). Irrational Number: An irrational number is a number that cannot be written in the form of \(\frac{p}{q}\), where q is not equal to zero and p and q are integers. Long Division Method: A method used for calculating the square roots of non-perfect squares by dividing step by step. Perfect Square: A number that is the square of an integer. For example, 16 is a perfect square because \(4^2 = 16\). Prime Number: A number greater than 1 that has no positive divisors other than 1 and itself. For example, 991 is a prime number.