Simplify the following expression:

\(\frac{6.5^3 -5^3}{42.25 + 32.5 +25}\)

Given:

Expression: \(\frac{6.5^3 -5^3}{42.25 + 32.5 +25}\)

Formula Used:

Algebraic Identity: a³ - b³ = (a - b)(a² + ab + b²)

Calculation:

Let a = 6.5 and b = 5.

Numerator: a³ - b³ = 6.5³ - 5³

Observe the terms in the denominator:

42.25 = 6.5² = a²

32.5 = 6.5 × 5 = ab

25 = 5² = b²

So, the denominator is a² + ab + b² = 6.5² + (6.5 × 5) + 5²

⇒ \(\frac{6.5^3 -5^3}{42.25 + 32.5 +25}\) = \(\frac{a^3 - b^3}{a^2 + ab + b^2}\) 

Using the identity a³ - b³ = (a - b)(a² + ab + b²):

\(\frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2}\)

Cancel out the common term (a² + ab + b²) from the numerator and denominator.

⇒ a - b

Substitute the values of a and b:

⇒ 6.5 - 5

⇒ 1.5

∴ The value of the expression is 1.5.