Square Root of 958

The square root is the inverse operation of squaring a number. Since 958 is not a perfect square, its square root is expressed in both radical and exponential form. In the radical form, it is expressed as √958, and in the exponential form, it is expressed as (958)^(1/2). The square root of 958 is approximately equal to 30.935, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method works well for perfect squares, but for non-perfect squares like 958, we use the long division method and approximation method. Let us now explore these methods:

• Prime factorization method
   
• Long division method
   
• Approximation method

The prime factorization of a number is the product of its prime factors. Let's see how 958 can be broken down into its prime factors:

Step 1: Finding the prime factors of 958. Breaking it down, we get 2 x 479. Since 479 is a prime number, there are no further factors.

Step 2: Now that we have the prime factors of 958, it is clear that they cannot be paired completely, as 958 is not a perfect square.

Therefore, calculating the square root of 958 using prime factorization is not feasible.

The long division method is suitable for finding the square root of non-perfect squares. Here is how to find the square root of 958 using this method, step by step:

Step 1: Group the numbers from right to left. For 958, we consider it as 9|58.

Step 2: Find a number whose square is less than or equal to 9. That number is 3, because 3 x 3 = 9. Subtracting gives a remainder of 0.

Step 3: Bring down 58 to make the new dividend 58. Double the previous quotient (3) to get 6, which becomes the beginning of the new divisor.

Step 4: Find a digit (n) such that 6n x n is less than or equal to 58. The suitable n is 0, because 60 x 0 = 0.

Step 5: Subtract 0 from 58, leaving 58.

Step 6: Since the dividend is less than the divisor, add a decimal point and two zeroes, making the new dividend 5800.

Step 7: Now find a digit (n) for the new divisor 600 + n such that (60n + n) x n is less than or equal to 5800. The suitable n is 9, because 609 x 9 = 5481.

Step 8: Subtract 5481 from 5800 to get a remainder of 319.

Step 9: The quotient is approximately 30.9.

Step 10: Continue this process to obtain more precision.

So, the square root of √958 is approximately 30.935.

The approximation method is another way to find square roots, especially when an exact value is not necessary. Here is how to approximate the square root of 958:

Step 1: Find the closest perfect squares around 958. The closest perfect square less than 958 is 961 (31²), and the closest perfect square greater is 900 (30²). Therefore, √958 falls between 30 and 31.

Step 2: Use linear interpolation to estimate the decimal part. Formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) Applying the formula: (958 - 900) / (961 - 900) = 58 / 61 ≈ 0.951

Step 3: Add this decimal to the smaller integer: 30 + 0.951 ≈ 30.951 Thus, the square root of 958 is approximately 30.951.

• Square root: A square root is the inverse of squaring a number. For example, 9^2 = 81, and the square root of 81 is √81 = 9.

• Irrational number: An irrational number cannot be expressed as a simple fraction. For example, the square root of 2 is an irrational number.

• Approximation: A method of finding a value that is close to, but not exactly, a precise value.

• Long division method: A technique to find the square root of a number by dividing it into smaller parts, making it easier to calculate.

• Prime factorization: Breaking down a composite number into its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.