Square Root of 34.96

The square root is the inverse of the square of the number. 34.96 is not a perfect square. The square root of 34.96 is expressed in both radical and exponential form. In the radical form, it is expressed as √34.96, whereas (34.96)^(1/2) in the exponential form. √34.96 = 5.912, which is a rational number because it can be expressed as a fraction.

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 34.96 is broken down into its prime factors:

Step 1: Convert 34.96 to a fraction, which is 3496/100.

Step 2: Find the prime factors of both the numerator and the denominator.

- Prime factors of 3496 are 2 x 2 x 2 x 2 x 3 x 73.

- Prime factors of 100 are 2 x 2 x 5 x 5.

Step 3: Since 34.96 is not a perfect square, finding the exact square root using prime factorization is complex.

The long division method is particularly used for non-perfect square numbers. In this method, we find the square root step by step.

Step 1: Group the numbers from right to left. Here, we treat it as 34.96.

Step 2: Find the largest number whose square is less than or equal to 34. The number is 5 (since 5 x 5 = 25).

Step 3: Subtract 25 from 34, giving a remainder of 9. Bring down 96, making it 996.

Step 4: Double the current quotient (5), which is 10. Now find a digit (n) such that 10n x n ≤ 996. The digit is 9 (since 109 x 9 = 981).

Step 5: Subtract 981 from 996, leaving a remainder of 15. Add decimal points and two zeros, making it 1500.

Step 6: Repeat this process to get the decimal places.

Continuing yields a quotient of 5.912.

The approximation method is another method for finding the 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 34.96 using the approximation method.

Step 1: Identify the closest perfect squares around 34.96. The closest perfect squares are 25 (5^2) and 36 (6^2).

Step 2: Since 34.96 is closer to 36 than to 25, we approximate the square root to be slightly less than 6.

Step 3: By checking between 5.9 and 6, we find the square root of 34.96 is approximately 5.912.

• 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, that is, √16 = 4.
   
• Rational number: A rational number is a number that can 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 why it is also known as the principal square root.
   
• Decimals: 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 1, 4, 9, 16, and 25 are perfect squares.