Square Root of 380

The square root is the inverse operation of squaring a number. 380 is not a perfect square. The square root of 380 can be expressed in both radical and exponential form. In radical form, it is expressed as √380, whereas in exponential form, it is (380)^(1/2). The square root of 380 is approximately 19.49359, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method is typically used for perfect square numbers. For non-perfect square numbers, methods such as the long division method and approximation method are used. Let's explore these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as a product of prime factors. Let's break down 380 into its prime factors:

Step 1: Finding the prime factors of 380 Breaking it down, we get 2 x 2 x 5 x 19: 2^2 x 5^1 x 19^1

Step 2: We have identified the prime factors of 380. Since 380 is not a perfect square, the digits cannot be grouped into pairs.

Thus, calculating √380 using prime factorization directly is not feasible.

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

Step 1: Begin by grouping the digits of 380 from right to left. In this case, it’s 80 and 3.

Step 2: Find the largest number whose square is less than or equal to 3. That number is 1 because 1^2 = 1. After subtracting, the remainder is 2.

Step 3: Bring down the next pair, 80, to make the new dividend 280. Double the quotient (1) to use as the next divisor, which becomes 2.

Step 4: Determine a digit 'n' such that 2n times n is less than or equal to 280. The number is 9, because 29 x 9 = 261.

Step 5: Subtract 261 from 280 to get a remainder of 19. The quotient is 19.

Step 6: Add a decimal point and bring down a pair of zeros to make the new dividend 1900.

Step 7: Double the current quotient (19) to get 38, which is part of the new divisor.

Step 8: Determine a digit 'n' such that 38n times n is less than or equal to 1900. The number is 4, because 384 x 4 = 1536.

Step 9: Subtract 1536 from 1900 to get a remainder of 364. The quotient is 19.4.

Step 10: Continue this process to obtain more decimal places.

Thus, the square root of 380 is approximately 19.493.

The approximation method is a simple way to find square roots. Here's how to find the square root of 380 using approximation:

Step 1: Identify the perfect squares closest to 380.

The closest are 361 (19^2) and 400 (20^2).

Therefore, √380 falls between 19 and 20.

Step 2: Use linear interpolation to approximate the root.

Calculate the proportion of 380 between 361 and 400.

(380 - 361) / (400 - 361) = 19 / 39 = 0.487

Adding this to 19 gives us 19.487.

Thus, the square root of 380 is approximately 19.487.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √25 = 5.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. It has a non-repeating, non-terminating decimal expansion.
   
• Principal square root: The non-negative square root of a number is called its principal square root. For example, the principal square root of 9 is 3.
   
• Prime factorization: Expressing a number as a product of its prime numbers. Example: 380 = 2² × 5 × 19.
   
• Long division method: A method used to calculate the square root of non-perfect squares by dividing the number into pairs and using iterative division.