Square Root of 7633

The square root is the inverse of the square of a number. 7633 is not a perfect square. The square root of 7633 is expressed in both radical and exponential forms. In radical form, it is expressed as √7633, whereas in exponential form it is expressed as (7633)^(1/2). √7633 ≈ 87.3603, 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:

• 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 7633 is broken down into its prime factors:

Step 1: Finding the prime factors of 7633. Breaking it down, we find that it is not straightforward to factorize 7633 as it is a large number and not a perfect square. Therefore, calculating 7633 using prime factorization is impractical for finding its square root.

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, group the numbers from right to left. In the case of 7633, we need to group it as 76 and 33.

Step 2: Find n whose square is less than or equal to 76. We can say n is 8 because 8 × 8 = 64, which is less than 76. The quotient is 8, and after subtracting, the remainder is 12.

Step 3: Bring down 33, making the new dividend 1233. Add the old divisor (8) with itself to get 16, which will be part of our new divisor.

Step 4: The new divisor will be 16n. Find the value of n such that 16n × n ≤ 1233. Let n = 7, so 167 × 7 = 1169.

Step 5: Subtract 1169 from 1233, the difference is 64, and the quotient is 87.

Step 6: Since the dividend is less than the divisor, we add a decimal point and two zeros to the dividend, making it 6400.

Step 7: Continue this process to obtain a more precise value. After several iterations, the approximate square root of 7633 is 87.360.

The approximation method is another method for finding square roots. It is a straightforward method to estimate the square root of a given number. Now let us learn how to find the square root of 7633 using the approximation method.

Step 1: Identify the closest perfect squares around 7633. The nearest perfect squares are 7569 (87^2) and 7744 (88^2). √7633 falls between 87 and 88.

Step 2: Apply interpolation to estimate the decimal. Use the formula: (Given number - Lower perfect square) / (Upper perfect square - Lower perfect square). (7633 - 7569) / (7744 - 7569) = 64 / 175 = 0.3657.

Step 3: Add this decimal to 87, giving 87 + 0.3657 ≈ 87.3657. Thus, the square root of 7633 is approximately 87.360.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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.
   
• Principal square root: A number has both positive and negative square roots, but it is always the positive square root that has more prominence due to its uses in the real world. This is known as the principal square root.
   
• Prime number: A prime number is a number greater than 1 that cannot be formed by multiplying two smaller natural numbers. It has only two factors: 1 and itself. For example, 7 is a prime number.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.