Square Root of 929

The square root is the inverse of the square of the number. 929 is not a perfect square. The square root of 929 is expressed in both radical and exponential form. In the radical form, it is expressed as √929, whereas 929^(1/2) is the exponential form. √929 ≈ 30.4795, 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 929 is broken down.

Step 1: Finding the prime factors of 929: 929 is a prime number, so it cannot be broken down into smaller prime factors.

Prime factorization is not applicable for 929 as it is not a composite number.

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 929, we need to group it as 29 and 9.

Step 2: Now we need to find a number whose square is closest to or less than 9. We can say n as ‘3’ because 3 × 3 = 9. Now the quotient is 3 after subtracting 9 from 9 the remainder is 0.

Step 3: Now let us bring down 29 which is the new dividend. Add the old divisor with the same number 3 + 3 we get 6 which will be our new divisor.

Step 4: The new divisor will be 6. We need to find 6n such that 6n × n ≤ 29. Let us consider n as 4, now 6 × 4 = 24.

Step 5: Subtract 24 from 29 the difference is 5, and the quotient is 34.

Step 6: 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 500.

Step 7: Now we need to find the new divisor that is 68 because 684 × 4 = 2736. We need to adjust n to get a closer approximation, let's take n as 7.

Step 8: Subtracting 476 from 500 we get the result 24.

Step 9: Now the quotient is 30.4.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √929 ≈ 30.48.

The 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 929 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √929.

The smallest perfect square less than 929 is 900 (30²) and the largest perfect square greater than 929 is 961 (31²).

√929 falls somewhere between 30 and 31.

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 (929 - 900) ÷ (961 - 900) = 29 / 61 ≈ 0.475

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 30 + 0.475 = 30.475, so the square root of 929 is approximately 30.475.

• Square root: A square root is the inverse of a square. Example: 4² = 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.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
   
• Approximation: An approximation is a value or number that is close to the exact value but not exact, used when exact values are difficult to obtain.