Square Root of 16000

The square root is the inverse of the square of a number. 16000 is not a perfect square. The square root of 16000 is expressed in both radical and exponential form. In the radical form, it is expressed as √16000, whereas (16000)^(1/2) in the exponential form. √16000 ≈ 126.4911, 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 typically used for non-perfect square numbers, where the long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 16000 is broken down into its prime factors.

Step 1: Finding the prime factors of 16000 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5: 2⁶ × 5⁴

Step 2: Now that we have found the prime factors of 16000, the next step is to make pairs of those prime factors. Since 16000 is not a perfect square, the digits of the number can’t be grouped entirely into pairs.

Therefore, calculating 16000 using prime factorization yields √(2⁶ × 5⁴) = 2³ × 5² × √(2 × 5) = 100 × √10.

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 group the numbers from right to left. In the case of 16000, we group it as 16 and 000.

Step 2: Now we need to find n whose square is 16. We can say n as '4' because 4 × 4 is equal to 16. Now the quotient is 4, after subtracting 16-16 the remainder is 0.

Step 3: Now let us bring down 000, which is the new dividend. Add the old divisor with the same number 4 + 4, we get 8 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 80n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 80n × n ≤ 000. Since the dividend is 000, we proceed with adding decimal places to continue the division.

Step 6: Adding decimal places allows us to bring down more zeroes. Continue the division process to find more decimal places in the quotient. So the square root of √16000 ≈ 126.491.

The approximation method is another way to find 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 16000 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √16000. The smallest perfect square less than 16000 is 14400 (120²) and the largest perfect square more than 16000 is 16900 (130²). √16000 falls somewhere between 120 and 130.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (16000 - 14400) / (16900 - 14400) = 1600 / 2500 = 0.64. Using this, we identify the decimal point of our square root.

The next step is adding the value we got initially to the decimal number which is 120 + 0.64 ≈ 126.49, so the square root of 16000 is approximately 126.49.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse of the square is the square root, √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, but we often consider only the positive square root, known as the principal square root.

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point, for example, 7.86, 8.65, and 9.42.

• Long division method: A method used to find the square root of a number by dividing it into manageable parts, useful for non-perfect squares.