Square Root of 7488

The square root is the inverse of squaring a number. 7488 is not a perfect square. The square root of 7488 can be expressed in both radical and exponential forms. In radical form, it is expressed as √7488, whereas in exponential form it is (7488)^(1/2). √7488 ≈ 86.521, which is an irrational number because it cannot be expressed as a ratio of two integers.

For perfect square numbers, the prime factorization method is used. However, for non-perfect square numbers like 7488, the long division method and approximation method are used. Let us now learn these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us see how 7488 is broken down into its prime factors:

Step 1: Finding the prime factors of 7488: Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 13 x 12.

Step 2: Now that we have found the prime factors of 7488, the next step is to make pairs of those prime factors. Since 7488 is not a perfect square, the digits of the number cannot be grouped in pairs. Therefore, calculating the square root of 7488 using prime factorization directly is not straightforward.

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 7488, we need to group it as 88 and 74.

Step 2: Now we need to find n whose square is less than or equal to 74. We can choose n as ‘8’ because 8 x 8 = 64 is less than 74. Now the quotient is 8, and after subtracting 64 from 74, the remainder is 10.

Step 3: Bring down 88, making the new dividend 1088. Add the old divisor 8 to itself to get 16, which will be part of our new divisor.

Step 4: The new divisor will be 16n. We need to find n such that 16n x n is less than or equal to 1088. Choosing n as 6, we find 16 x 6 x 6 = 576.

Step 5: Subtract 576 from 1088 to get 512, and the quotient becomes 86.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point, allowing us to bring down two zeros. The new dividend becomes 51200.

Step 7: Now we find a new divisor, 172, such that 172n x n is less than or equal to 51200. Choosing n as 2, we find 172 x 2 x 2 = 688.

Step 8: Subtract 688 from 51200 to get 50512, and continue this process until the desired accuracy is achieved. So the square root of √7488 is approximately 86.52.

The approximation method is another way to find square roots and is an easy method for estimating the square root of a given number. Let us learn how to find the square root of 7488 using the approximation method.

Step 1: First, find the closest perfect squares to 7488. The closest perfect squares are 7225 (85^2) and 7569 (87^2). √7488 falls between 85 and 87.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula (7488 - 7225) / (7569 - 7225) = 263 / 344 = 0.7645.

Step 3: Adding this to the smaller perfect square root gives 85 + 0.7645 ≈ 85.76. So, the approximate square root of 7488 using this method is about 85.76.

• Square root: The square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse is the square root: √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction, meaning it cannot be written as p/q where p and q are integers and q ≠ 0.
   
• Principal square root: A number has both positive and negative square roots, but the positive square root is commonly used in practical applications and is known as the principal square root.
   
• Prime factorization: Breaking down a number into the product of its prime factors. For example, 7488 can be expressed as 2 x 2 x 2 x 2 x 3 x 3 x 13 x 12.
   
• Long division method: A method used to find the square root of a number, particularly useful for non-perfect squares, involving several systematic steps.