Square Root of 1159

The square root is the inverse of the square of the number. 1159 is not a perfect square. The square root of 1159 is expressed in both radical and exponential form. In the radical form, it is expressed as √1159, whereas (1159)^(1/2) in exponential form. √1159 ≈ 34.0454, 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, the prime factorization method is not applicable. Instead, long-division and approximation methods 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 1159 is broken down into its prime factors.

Step 1: Finding the prime factors of 1159 Breaking it down, we get 7 × 13 × 127. Since 1159 is not a perfect square, these factors cannot be paired. Therefore, calculating 1159 using prime factorization as a perfect square is impossible.

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: Group the numbers from right to left. For 1159, group it as 59 and 11.

Step 2: Find n whose square is ≤ 11. We can say n as ‘3’ because 3 × 3 = 9, which is less than 11. Now the quotient is 3, and the remainder is 11 - 9 = 2.

Step 3: Bring down 59 to make the new dividend 259.

Step 4: Double the quotient and use it as the new divisor's first digit: 3 × 2 = 6.

Step 5: Now find a digit x such that 6x × x is less than or equal to 259. Try 4: 64 × 4 = 256.

Step 6: Subtract 256 from 259 to get the remainder 3.

Step 7: Since the remainder is less than the divisor, add a decimal point and two zeroes to continue. The new dividend is 300.

Step 8: Repeat the process to find more decimal places. So the square root of √1159 ≈ 34.0454.

The approximation method is another approach for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 1159 using the approximation method.

Step 1: Find the closest perfect squares of √1159. The smallest perfect square less than 1159 is 1156 (34²) and the largest perfect square more than 1159 is 1225 (35²). √1159 falls between 34 and 35.

Step 2: Use interpolation to approximate: (1159 - 1156) / (1225 - 1156) = 3 / 69 ≈ 0.0435. Adding this to the lower bound: 34 + 0.0435 ≈ 34.0454. Therefore, the square root of 1159 is approximately 34.0454.

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

• Irrational number: An irrational number cannot be expressed as a simple fraction, where p and q are integers and q is not zero. Examples include √2 and √1159.

• Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 16 is 4, not -4.

• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 1159 is 7 × 13 × 127.

• Approximation: Estimating a value to a certain degree of accuracy, such as finding √1159 ≈ 34.0454.