Solve Algebraic Expression Using Identities

The problem requires simplifying the given expression:

$ \frac{0.6 \times 0.6 \times 0.6 + 0.7 \times 0.7 \times 0.7}{0.6 \times 0.6 - 0.42 + 0.7 \times 0.7} $

Let's identify the structure. We can see terms like $0.6^3$ and $0.7^3$ in the numerator and terms related to $0.6^2$ and $0.7^2$ in the denominator.

Applying Algebraic Identity

We can use the algebraic identity for the sum of cubes:

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

Let $a = 0.6$ and $b = 0.7$.

• The numerator is $a^3 + b^3$.
• The denominator is $a^2 - (0.6 \times 0.7) + b^2$. Since $0.6 \times 0.7 = 0.42$, the denominator is $a^2 - ab + b^2$.

Substituting these into the expression, we get:

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

Using the identity, this simplifies to:

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

Final Calculation

Now, substitute the values of $a$ and $b$ back:

$ a + b = 0.6 + 0.7 $

$ 0.6 + 0.7 = 1.3 $

Therefore, the value of the expression is 1.3.