Square Root of 1445

The square root is the inverse of the square of the number. 1445 is not a perfect square. The square root of 1445 is expressed in both radical and exponential form. In radical form, it is expressed as √1445, whereas in exponential form, it is (1445)^(1/2). √1445 ≈ 37.9987, 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, for non-perfect square numbers like 1445, 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 1445 is broken down into its prime factors.

Step 1: Finding the prime factors of 1445 Breaking it down, we get 5 x 17 x 17: 5^1 x 17^2

Step 2: Now we have found the prime factors of 1445. The second step is to make pairs of those prime factors. Since 1445 is not a perfect square, the digits of the number can’t be grouped in complete pairs.

Therefore, calculating 1445 using prime factorization alone isn't 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 1445, we need to group it as 45 and 14.

Step 2: Now we need to find n whose square is less than or equal to 14. We can say n is ‘3’ because 3 x 3 = 9 is less than or equal to 14. Now the quotient is 3; after subtracting 9 from 14, the remainder is 5.

Step 3: Now let us bring down 45, which is the new dividend. Add the old divisor with the same number, 3 + 3, to get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we have 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 545. Let us consider n as 9, now 69 x 9 = 621.

Step 6: Subtract 545 from 621; the difference is negative, so we need to try n as 8. So, 68 x 8 = 544.

Step 7: Subtracting 544 from 545, the difference is 1, and the quotient is 38.

Step 8: Since the remainder is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 100.

Step 9: The new divisor is 760 because 760 x 1 = 760.

Step 10: Subtracting 760 from 1000, we get a result of 240.

Step 11: Continue these steps until we get two decimal places.

So the square root of √1445 ≈ 37.998

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 1445 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1445.

The smallest perfect square less than 1445 is 1444, and the largest perfect square greater than 1445 is 1521. √1445 falls somewhere between 38 and 39.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square)/(Next perfect square - smallest perfect square)

Using the formula (1445 - 1444) / (1521 - 1444) = 1/77 ≈ 0.01298 Adding this to the lower bound, we get 38 + 0.01298 = 38.01298.

So the square root of 1445 is approximately 38.01.

• Square root: A square root is the inverse of squaring a number. For example: 4^2 = 16, and the inverse of the square is the square root, which is √16 = 4.

• Irrational number: An irrational number is one 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 prominent due to its uses in the real world. This is known as the principal square root.

• Prime factorization: Breaking down a number into its basic prime number multipliers is known as prime factorization. For example, for 1445, it is 5 × 17 × 17.

• Long division method: A step-by-step method used to find the square root of non-perfect squares by grouping numbers and determining the closest possible root.