Square Root of 390

The square root is the inverse of squaring a number. 390 is not a perfect square. The square root of 390 can be expressed in both radical and exponential forms. In radical form, it is expressed as √390, whereas in exponential form, it is (390)^(1/2). √390 ≈ 19.748, which is an irrational number because it cannot be expressed as a fraction 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 390, the long division method and approximation method are more appropriate. Let us explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Let's see how 390 is broken down into its prime factors.

Step 1: Finding the prime factors of 390 Breaking it down, we get 2 x 3 x 5 x 13: 2^1 x 3^1 x 5^1 x 13^1

Step 2: We found the prime factors of 390. The next step is to make pairs of those prime factors. Since 390 is not a perfect square, the digits of the number cannot be grouped into pairs, making it impossible to calculate the square root using prime factorization.

The long division method is particularly useful for non-perfect square numbers. Let's learn how to find the square root using the long division method, step by step.

Step 1: To begin, group the numbers from right to left. In the case of 390, we group it as 90 and 3.

Step 2: Find n whose square is less than or equal to 3. We choose n = 1 because 1 x 1 ≤ 3. The quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 90 to get a new dividend of 290. Add the old divisor (1) to itself to get 2, the new divisor.

Step 4: Find n such that 2n x n ≤ 290. If n = 9, 29 x 9 = 261.

Step 5: Subtract 261 from 290 to get a remainder of 29

Step 6: Add a decimal point and bring down two zeros to make the new dividend 2900.

Step 7: Find the new divisor. Adding a digit to 58 yields 589. Choose n = 4, resulting in 589 x 4 = 2356.

Step 8: Subtract 2356 from 2900 to get 544.

Step 9: Continue the process until you reach the desired level of precision. The quotient is approximately 19.74.

So, the square root of √390 is approximately 19.74.

The approximation method is another useful approach to finding square roots. It is an easy way to estimate the square root of a number. Let's learn how to find the square root of 390 using the approximation method.

Step 1: Identify the closest perfect squares around 390.

The smallest perfect square less than 390 is 361, and the largest perfect square greater than 390 is 400.

√390 falls between 19 and 20.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

(390 - 361) / (400 - 361) ≈ 0.7436

Adding this decimal to the lower whole number, 19 + 0.7436 ≈ 19.74.

So, the approximate square root of 390 is 19.74.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root, so √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction, where p and q are integers, and q is not equal to zero.
   
• Principal square root: A number has both positive and negative square roots, but the positive square root is typically used in real-world applications, known as the principal square root.
   
• Factorization: Breaking down a number into its basic components, or factors, that, when multiplied together, give the original number.
   
• Approximation: Estimating a value that is close to the actual value, often used for numbers that cannot be expressed exactly, like irrational numbers.