Square Root of 525

The square root is the inverse of squaring a number. 525 is not a perfect square. The square root of 525 can be expressed in both radical and exponential forms. In radical form, it is expressed as √525, whereas in exponential form, it is (525)^(1/2). √525 ≈ 22.91288, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect squares. However, for non-perfect squares, methods like the long division method and approximation method are used. Let us explore these methods: -

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

Prime factorization involves expressing a number as a product of its prime factors. Let us break down 525 into its prime factors:

Step 1: Find the prime factors of 525 Breaking it down, we get 3 × 5 × 5 × 7: 3¹ × 5² × 7¹

Step 2: Since 525 is not a perfect square, the digits cannot be grouped into pairs.

Therefore, calculating the square root of 525 using prime factorization alone is not feasible.

The long division method is useful for finding the square root of non-perfect squares. Let's find the square root of 525 using this method, step by step:

Step 1: Group the digits of 525 from right to left as 25 and 5.

Step 2: Find the largest integer n whose square is less than or equal to 5. Here, n is 2 because 2 × 2 = 4. The remainder is 1.

Step 3: Bring down the next pair, 25, making the new dividend 125. Add the previous divisor to itself to get 4, making the new divisor.

Step 4: Determine n such that 4n × n ≤ 125. If n is 2, then 42 × 2 = 84.

Step 5: Subtract 84 from 125, giving a remainder of 41. The quotient so far is 22. Step 6: Bring down two zeros, making the new dividend 4100.

Step 7: Find the new divisor, which is 229, because 229 × 9 = 2061.

Step 8: Subtract 2061 from 4100, giving a remainder of 2039.

Step 9: The quotient so far is 22.9. Repeat these steps until you have sufficient decimal places.

Thus, √525 ≈ 22.91.

The approximation method is another approach to find square roots. Let's approximate the square root of 525:

Step 1: Identify the perfect squares closest to 525. 484 and 529 are the nearest perfect squares, with square roots 22 and 23, respectively.

Step 2: Apply the formula:

(Given number - smaller perfect square) / (Larger perfect square - smaller perfect square)

(525 - 484) / (529 - 484) = 41 / 45 ≈ 0.9111

Add this decimal to the smaller root: 22 + 0.9111 = 22.9111

Therefore, √525 ≈ 22.9111.

• Square root: The square root is the inverse of squaring a number. For example, 4² = 16, so the square root is √16 = 4.

• Irrational number: An irrational number cannot be expressed as a fraction of two integers, with a non-zero denominator. √525 is an example.

• Principal square root: The positive square root, often used in practical applications, is known as the principal square root. For example, the principal square root of 16 is 4.

• Prime factorization: This is the expression of a number as a product of its prime factors. For example, the prime factorization of 525 is 3 × 5² × 7.

• Long division method: A method for finding square roots of non-perfect squares by dividing in steps, offering a precise decimal value.