Square Root of 7048

The square root is the inverse of the square of the number. 7048 is not a perfect square. The square root of 7048 is expressed in both radical and exponential forms. In radical form, it is expressed as √7048, whereas (7048)^(1/2) in exponential form. √7048 ≈ 83.9452, 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 7048 is broken down into its prime factors.

Step 1: Finding the prime factors of 7048 Breaking it down, we get 2 × 2 × 2 × 2 × 19 × 37: 2^4 × 19 × 37

Step 2: Now we found out the prime factors of 7048. The second step is to make pairs of those prime factors. Since 7048 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 7048 using prime factorization is not feasible.

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 7048, we need to group it as 48 and 70.

Step 2: Now we need to find n whose square is less than or equal to 70. We can say n as ‘8’ because 8 × 8 = 64 is less than or equal to 70. Now, the quotient is 8; after subtracting 64 from 70, the remainder is 6.

Step 3: Now let us bring down 48, which is the new dividend. Add the old divisor with the same number 8 + 8 to get 16, which will be our new divisor.

Step 4: The new divisor will be 16n. We need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 648. Let us consider n as 4, now 16 × 4 = 64.

Step 6: Subtract 648 from 640, the difference is 8, and the quotient is 84.

Step 7: 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 800.

Step 8: Now we need to find the new divisor, which is 168 because 168 × 4 = 672.

Step 9: Subtracting 672 from 800, we get the result 128.

Step 10: Now the quotient is 83.9

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √7048 ≈ 83.94

The approximation method is another method for finding square roots. 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 7048 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7048. The smallest perfect square below 7048 is 6889, and the largest perfect square above 7048 is 7225. √7048 falls somewhere between 83 and 85.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula: (7048 - 6889) ÷ (7225 - 6889) ≈ 0.945 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 83 + 0.945 ≈ 83.945. So the square root of 7048 is approximately 83.945.

• 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 that 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; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: Expressing a number as the product of its prime numbers.
   
• 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.