Square Root of 7.5

The square root is the inverse of the square of the number. 7.5 is not a perfect square. The square root of 7.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √7.5, whereas (7.5)^(1/2) in the exponential form. √7.5 = 2.73861, 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 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 7.5 is broken down into its prime factors.

Step 1: Express 7.5 as a fraction, 15/2, to find its prime factors.

Step 2: The prime factors of 15 are 3 × 5, and the prime factors of 2 are 2 itself.

Step 3: Therefore, the prime factorization of 7.5 is 3 × 5 × 2^-1. Since 7.5 is not a perfect square, a meaningful pair cannot be formed for the square root.

Therefore, calculating the square root of 7.5 using prime factorization 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 numbers for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Start by grouping the digits of 7.5 as 75 and 0 to the right of the decimal.

Step 2: Find n such that n² is less than or equal to 7. The closest value is 2, since 2² = 4. The quotient is 2, and the remainder is 3 after subtracting 4 from 7.

Step 3: Bring down the next pair of digits (50 in this case) to make it 350.

Step 4: Double the quotient (2) to get 4, which will be our new divisor. We need to find n such that 4n × n ≤ 350.

Step 5: The value of n is 7, since 47 × 7 = 329. Subtract 329 from 350 to get a remainder of 21. Append two zeros to get 2100.

Step 6: Continue the process until you achieve sufficient decimal places. For √7.5, the quotient will start with 2.738.

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 7.5 using the approximation method.

Step 1: Find the closest perfect squares around 7.5.

The closest perfect squares are 4 (2²) and 9 (3²). √7.5 falls between 2 and 3.

Step 2: Apply the formula (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using this, (7.5 - 4) / (9 - 4) = 3.5 / 5 = 0.7.

Step 3: Add this decimal to the smaller square root value, giving 2 + 0.7 = 2.7 as an approximation. Refining further, we find 2.73861 as a more precise value.

• 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.

• 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.

• Fraction: A fraction represents a part of a whole or, more generally, any number of equal parts. Example: 7.5 can be expressed as a fraction 15/2.

• Principal square root: A number has both positive and negative square roots. However, the positive square root is more prominent due to its uses in the real world. That is the reason it is also known as a principal square root.