Square Root of 43000

The square root is the inverse of the square of the number. 43000 is not a perfect square. The square root of 43000 is expressed in both radical and exponential form. In the radical form, it is expressed as √43000, whereas (43000)^(1/2) in the exponential form. √43000 ≈ 207.364, 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 43000 is broken down into its prime factors.

Step 1: Finding the prime factors of 43000 Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 5 x 43: 2^3 x 5^3 x 43

Step 2: Now we found out the prime factors of 43000. The second step is to make pairs of those prime factors. Since 43000 is not a perfect square, therefore the digits of the number can’t be grouped in pair.

Therefore, calculating 43000 using prime factorization is impossible.

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 43000, we need to group it as 00 and 430.

Step 2: Now we need to find n whose square is less than or equal to 430. We can say n as ‘20’ because 20 x 20 = 400 is lesser than 430. Now the quotient is 20 after subtracting 400 from 430, the remainder is 30.

Step 3: Now let us bring down 00 which is the new dividend. Add the old divisor with the same number, 20 + 20, we get 40 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 40n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 40n × n ≤ 3000. Let us consider n as 7, now 407 x 7 = 2849.

Step 6: Subtract 2849 from 3000, the difference is 151, and the quotient is 207.

Step 7: 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 15100.

Step 8: Now we need to find the new divisor that is 414 because 414 x 3 = 1242.

Step 9: Subtracting 1242 from 15100 we get the result 25858.

Step 10: Now the quotient is 207.3.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √43000 ≈ 207.36.

Approximation method is another method for finding the 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 43000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √43000. The smallest perfect square less than 43000 is 42250, and the largest perfect square more than 43000 is 43560. √43000 falls somewhere between 205 and 210.

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

Going by the formula (43000 - 42250) ÷ (43560 - 42250) = 0.75.

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 which is 205 + 0.75 = 205.75, so the square root of 43000 is approximately 205.75.

• 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 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; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.

• Approximation: Estimating a value based on logical reasoning and closest values, typically used to find square roots of non-perfect squares.