Square Root of 345

The square root is the inverse of the square of the number. 345 is not a perfect square. The square root of 345 is expressed in both radical and exponential form. In radical form, it is expressed as √345, whereas (345)^(1/2) is the exponential form. √345 ≈ 18.5742, 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 345 is broken down into its prime factors:

Step 1: Finding the prime factors of 345 Breaking it down, we get 3 x 5 x 23: 3^1 x 5^1 x 23^1

Step 2: Now we found out the prime factors of 345. The second step is to make pairs of those prime factors. Since 345 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 345 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 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 345, we need to group it as 45 and 3.

Step 2: Now we need to find n whose square is closest to 3. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 45, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be 2n, and we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 245. Let us consider n as 9; now, 29 x 9 = 261.

Step 6: Since 261 is greater than 245, we use n as 8. Now 28 x 8 = 224.

Step 7: Subtract 224 from 245; the difference is 21. The quotient is now 18.

Step 8: 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 2100.

Step 9: Now we need to find the new divisor. We get 369 when 369 x 5 = 1845.

Step 10: Subtracting 1845 from 2100, we get the result 255.

Step 11: Now the quotient is 18.5.

Step 12: Continue doing these steps until we get two numbers after the decimal point.

The square root of √345 ≈ 18.57.

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 345 using the approximation method.

Step 1: Now we have to find the closest perfect square of √345.

The smallest perfect square less than 345 is 324, and the largest perfect square greater than 345 is 361. √345 falls somewhere between 18 and 19.

Step 2: Now we need to apply the formula that is: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square) Going by the formula (345 - 324) ÷ (361 - 324) = 0.567

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: 18 + 0.567 = 18.567, so the square root of 345 is approximately 18.57.

• 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.

• Irrational number: An irrational number is a number that cannot be written in the form 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 often the positive square root that is emphasized due to its usefulness in real-world applications. This is why it is also known as the principal square root.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, 345 = 3 x 5 x 23.

• Decimal: A number that includes a whole number and a fractional part, separated by a decimal point. For example: 7.86, 8.65, and 9.42 are decimal numbers.