Square Root of 7.05

The square root is the inverse of the square of a number. 7.05 is not a perfect square. The square root of 7.05 is expressed in both radical and exponential forms. In radical form, it is expressed as √7.05, whereas in exponential form it is (7.05)^(1/2). √7.05 ≈ 2.655, 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 prime factorization of a number is the product of its prime factors. However, since 7.05 is not a perfect square, it cannot be exactly broken down into pairs of prime factors to find its square root using this method. Therefore, calculating √7.05 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, group the numbers from right to left. In the case of 7.05, we treat it as 705.

Step 2: Find the largest number whose square is less than or equal to 7. The closest such number is 2, because 2 × 2 = 4.

Step 3: Subtract 4 from 7, getting a remainder of 3. Bring down the next pair of digits (05) to get 305.

Step 4: Double the divisor (2) to get 4. Now, find a number n such that 4n × n ≤ 305. The closest number is 6, because 46 × 6 = 276.

Step 5: Subtract 276 from 305, the remainder is 29.

Step 6: Add a decimal point and bring down 00 to get 2900.

Step 7: The new divisor is 52. Find a number n such that 52n × n ≤ 2900. The closest number is 5, because 525 × 5 = 2625.

Step 8: Subtract 2625 from 2900, getting a remainder of 275.

Step 9: The quotient so far is 2.65. Continue these steps until you get the desired precision.

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 7.05 using the approximation method.

Step 1: Identify the closest perfect squares around 7.05.

The closest perfect squares are 4 (√4 = 2) and 9 (√9 = 3).

Step 2: 7.05 is closer to 4 than to 9. Its square root will be closer to 2.

Step 3: Use interpolation or estimation to refine the approximation: (7.05 - 4) / (9 - 4) ≈ 0.61.

Step 4: The approximation of √7.05 ≈ 2 + 0.61 = 2.61. Continue refining the approximation for greater accuracy.

• Square root: A square root is the inverse of a square. Example: 4² = 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, but the positive square root is more commonly used in real-world applications, known as the principal square root.

• Long division method: A technique used to find square roots of non-perfect squares by dividing the number into groups and solving step-by-step.

• Approximation method: A technique for estimating the square root of a number by comparing it with nearby perfect squares and refining the result through interpolation.