Finding the Value of the Square Root of 2025

The question asks us to find the value of \(\sqrt{2025}\). Finding the square root of a number means finding a number that, when multiplied by itself, gives the original number. In this case, we need to find a number \(x\) such that \(x \times x = 2025\), or \(x^2 = 2025\).

Understanding Square Roots

A square root is the inverse operation of squaring a number. For example, the square root of 25 is 5 because \(5 \times 5 = 25\).

Methods to Find the Square Root of 2025

There are several ways to find the square root of a number like 2025. Two common methods are the prime factorization method and the long division method. We will use the prime factorization method as 2025 ends in 5, which suggests it is easily divisible by 5.

Prime Factorization Method for \(\sqrt{2025}\)

Step 1: Find the prime factors of 2025.

• 2025 is divisible by 5 (since it ends in 5). \(2025 \div 5 = 405\).
• 405 is divisible by 5. \(405 \div 5 = 81\).
• 81 is divisible by 3. \(81 \div 3 = 27\).
• 27 is divisible by 3. \(27 \div 3 = 9\).
• 9 is divisible by 3. \(9 \div 3 = 3\).
• 3 is a prime number.

So, the prime factorization of 2025 is \(3 \times 3 \times 3 \times 3 \times 5 \times 5\).

We can write this using exponents: \(2025 = 3^4 \times 5^2\).

Step 2: Group the prime factors into pairs.

To find the square root, we group identical prime factors into pairs:

\(2025 = (3 \times 3) \times (3 \times 3) \times (5 \times 5)\)

This can be written as:

\(2025 = 3^2 \times 3^2 \times 5^2\)

Step 3: Take one factor from each pair.

The square root of a number is found by taking one factor from each pair of prime factors:

\(\sqrt{2025} = \sqrt{3^2 \times 3^2 \times 5^2}\)

Using the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) and \(\sqrt{x^2} = x\):

\(\sqrt{2025} = \sqrt{3^2} \times \sqrt{3^2} \times \sqrt{5^2}\)

\(\sqrt{2025} = 3 \times 3 \times 5\)

Step 4: Multiply the factors obtained in Step 3.

\(\sqrt{2025} = 3 \times 3 \times 5 = 9 \times 5 = 45\)

So, the value of \(\sqrt{2025}\) is 45.

Verification

To verify the answer, we can square 45:

\(45 \times 45 = 45^2\)

\(45 \times 45 = (40 + 5) \times (40 + 5)\)

\(= 40 \times 40 + 40 \times 5 + 5 \times 40 + 5 \times 5\)

\(= 1600 + 200 + 200 + 25\)

\(= 1600 + 400 + 25\)

\(= 2000 + 25\)

\(= 2025\)

Since \(45^2 = 2025\), our calculation that \(\sqrt{2025} = 45\) is correct.

Comparing with Options

Let's check the given options:

• Option 1: 55. \(55^2 = 55 \times 55 = 3025\). Incorrect.
• Option 2: 25. \(25^2 = 25 \times 25 = 625\). Incorrect.
• Option 3: 65. \(65^2 = 65 \times 65 = 4225\). Incorrect.
• Option 4: 45. \(45^2 = 45 \times 45 = 2025\). Correct.

The value of \(\sqrt{2025}\) is 45.

Revision Table: Square Roots

Additional Information: Properties of Square Roots

Understanding the properties of square roots can help in solving problems involving square roots.

• Product Property: The square root of a product is the product of the square roots: \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\). This property was used in the prime factorization method.
• Quotient Property: The square root of a quotient is the quotient of the square roots: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) (where \(b \ne 0\)).
• Squaring a Square Root: Squaring a square root of a non-negative number gives the original number: \((\sqrt{x})^2 = x\) (where \(x \ge 0\)).
• Square Root of a Square: The square root of a square of a number is the absolute value of the number: \(\sqrt{x^2} = |x|\). For non-negative numbers, this is simply \(x\). In \(\sqrt{2025}\), we are looking for the principal (positive) square root.

These properties are fundamental when working with expressions involving square roots and exponents.