Simplify the Algebraic Fraction

The question asks us to simplify the following mathematical expression:

$$ \frac{2.5 \times 2.5 \times 2.5 - 1.5 \times 1.5 \times 1.5}{2.5 \times 2.5 + 2.5 \times 1.5 + 1.5 \times 1.5} $$

This expression can be written more compactly using exponents:

$$ \frac{2.5^3 - 1.5^3}{2.5^2 + (2.5 \times 1.5) + 1.5^2} $$

Using the Difference of Cubes Identity

To simplify this fraction, we can recognise that it fits a standard algebraic identity. The identity for the difference of two cubes is:

$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$

Let's compare this identity to our given expression. We can identify:

• Let \( a = 2.5 \)
• Let \( b = 1.5 \)

If we substitute these values into the identity, we get:

$$ (2.5)^3 - (1.5)^3 = (2.5 - 1.5)((2.5)^2 + (2.5 \times 1.5) + (1.5)^2) $$

Our original expression is the fraction \( \frac{2.5^3 - 1.5^3}{2.5^2 + (2.5 \times 1.5) + 1.5^2} \). Using our variables \(a\) and \(b\), this can be written as:

$$ \text{Expression} = \frac{a^3 - b^3}{a^2 + ab + b^2} $$

Simplifying the Fraction

Now, we can substitute the factored form of \( a^3 - b^3 \) from the identity into our expression:

$$ \text{Expression} = \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} $$

We can see that the term \( (a^2 + ab + b^2) \) exists in both the numerator and the denominator. Since \( a = 2.5 \) and \( b = 1.5 \), both are positive numbers. This means \( a^2 + ab + b^2 \) will be a positive value and certainly not zero. Therefore, we can safely cancel out this common factor:

$$ \text{Expression} = a - b $$

Calculating the Final Result

The expression simplifies to \( a - b \). Now, we substitute the values of \(a\) and \(b\) back into this simplified form:

$$ \text{Result} = 2.5 - 1.5 $$

Performing the subtraction:

$$ \text{Result} = 1 $$

So, the simplified value of the given expression is 1.

Comparing with Options

The final calculated result is 1. Let's check this against the provided options:

• Option 1: 2.5
• Option 2: 1.5
• Option 3: 1
• Option 4: \( (2.5)^2 - (1.5)^2 \)

Our calculation confirms that Option 3 is the correct answer.