Square Root of 353

The square root is the inverse of the square of the number. 353 is not a perfect square. The square root of 353 is expressed in both radical and exponential form. In the radical form, it is expressed as √353, whereas (353)^(1/2) in the exponential form. √353 ≈ 18.79, 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 353 is broken down into its prime factors:

Step 1: Finding the prime factors of 353 Breaking it down, we get 353 itself as 353 is a prime number.

Step 2: Now we found out the prime factors of 353. Since 353 is not a perfect square, calculating 353 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 353, we need to group it as 53 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can say n is ‘1’ because 1 × 1 = 1 which is less than or equal to 3. Now the quotient is 1, after subtracting 3-1, the remainder is 2.

Step 3: Now let us bring down 53 which forms the new dividend 253. Add the old divisor with the same number 1 + 1 to get 2 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 253. Let us consider n as 9, now 29 × 9 = 261, which is greater than 253. So we try n as 8, now 28 × 8 = 224.

Step 6: Subtract 253 from 224, the difference is 29, and the quotient is 18.

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 zeros to the dividend. Now the new dividend is 2900.

Step 8: Now we need to find the new divisor that is 189 because 189 × 9 = 1701.

Step 9: Subtracting 1701 from 2900 gives the result 1199.

Step 10: Now the quotient is 18.7.

Step 11: Continue these steps until the desired decimal places are achieved.

So the square root of √353 ≈ 18.79.

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 353 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √353.

The smallest perfect square below 353 is 324, and the largest perfect square above 353 is 361. √353 falls somewhere between 18 and 19.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Going by the formula (353 - 324) ÷ (361 - 324) = 29 ÷ 37 ≈ 0.7838.

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 18 + 0.7838 ≈ 18.79, so the square root of 353 is approximately 18.79.

• 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; however, it is always the positive square root that has more prominence due to its uses in the real world. This is why it is also known as the principal square root.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, etc., are prime numbers.

• Long division method: A method used to find the square root of a number that is not a perfect square, involving several steps of division and subtraction to achieve a precise value.