Simplify Expression: Detailed Solution

To simplify the given mathematical expression, we need to evaluate the numerator and the denominator separately and then perform the division. The expression is:

\( \frac{7.35 \times 7.35 - 2.25 \times 2.25}{0.24} \)

Let's look at the numerator: \( 7.35 \times 7.35 - 2.25 \times 2.25 \). This can be written as \( (7.35)^2 - (2.25)^2 \). This is in the form of a difference of squares, \( a^2 - b^2 \), where \( a = 7.35 \) and \( b = 2.25 \).

The difference of squares formula states that \( a^2 - b^2 = (a+b)(a-b) \). Using this formula for the numerator:

\( (7.35)^2 - (2.25)^2 = (7.35 + 2.25)(7.35 - 2.25) \)

First, calculate the sum \( (7.35 + 2.25) \):

\( 7.35 + 2.25 = 9.60 \)

Next, calculate the difference \( (7.35 - 2.25) \):

\( 7.35 - 2.25 = 5.10 \)

Now, substitute these values back into the factored numerator:

\( (9.60) \times (5.10) = 9.6 \times 5.1 \)

Let's multiply \( 9.6 \) by \( 5.1 \):

So, \( 9.6 \times 5.1 = 48.96 \). The numerator simplifies to \( 48.96 \).

Now, the expression becomes:

\( \frac{48.96}{0.24} \)

To perform this division, we can remove the decimal points by multiplying both the numerator and the denominator by 100:

\( \frac{48.96 \times 100}{0.24 \times 100} = \frac{4896}{24} \)

Now we need to divide 4896 by 24. We can do this using long division or by breaking down the numbers.

Using long division:

   204
  ____
24|4896
   -48
   ---
    09
    -0
    ---
     96
    -96
    ---
      0

Alternatively, we can see that \( 4800 \div 24 = 200 \) and \( 96 \div 24 = 4 \). So, \( \frac{4896}{24} = \frac{4800 + 96}{24} = \frac{4800}{24} + \frac{96}{24} = 200 + 4 = 204 \).

Therefore, the simplified value of the expression is 204.

Let's check the options:

• Option 1: 320
• Option 2: 225
• Option 3: 204
• Option 4: 304

Our calculated value, 204, matches Option 3.

Revision Table: Key Steps for Simplifying Expressions

Additional Information: Difference of Squares Identity

The difference of squares is a fundamental algebraic identity used to factor expressions. It states that the difference between the squares of two terms is equal to the product of the sum of the terms and the difference of the terms.

The formula is: \( a^2 - b^2 = (a+b)(a-b) \)

This identity is very useful in simplifying expressions, solving equations, and factoring polynomials. It allows us to break down a complex expression into simpler factors, which can make calculations easier, especially when dealing with decimals or fractions.

In this problem, applying the difference of squares formula simplified the numerator into a product of two numbers that were easier to work with than the original squares.