Square Root of 535

The square root is the inverse of the square of the number. 535 is not a perfect square. The square root of 535 is expressed in both radical and exponential form. In radical form, it is expressed as √535, whereas 535^1/2 in exponential form. √535 ≈ 23.151673, 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 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 535 is broken down into its prime factors.

Step 1: Finding the prime factors of 535 Breaking it down, we get 5 × 107: 5¹ × 107¹.

Step 2: Now we found out the prime factors of 535. The second step is to make pairs of those prime factors. Since 535 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 535 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 535, we need to group it as 35 and 5.

Step 2: Now we need to find n whose square is ≤ 5. We can say n as ‘2’ because 2 × 2 is 4, which is lesser than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 35, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n × n ≤ 135. Let us consider n as 3, now 43 × 3 = 129.

Step 5: Subtract 129 from 135; the difference is 6. The quotient is 23.

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 600.

Step 7: Now we need to find the new divisor that is 463, because 463 × 1 = 463.

Step 8: Subtracting 463 from 600, we get the result 137.

Step 9: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √535 is approximately 23.15.

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 535 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √535. The smallest perfect square less than 535 is 529, and the largest perfect square greater than 535 is 576. √535 falls somewhere between 23 and 24.

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 (535 - 529) ÷ (576 - 529) = 6 ÷ 47 ≈ 0.1277. 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 23 + 0.1277 ≈ 23.15, so the square root of 535 is approximately 23.15.

• 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.

• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

• 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.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 16 is a perfect square because it is 4².

• Long division method: A method used to divide numbers and find square roots of non-perfect squares by using a step-by-step division process.