Let’s break down the given expression step-by-step to show why the given result is:
[tex]\[
\frac{0.85 \cdot 0.85 \cdot 0.85 + 0.15 \cdot 0.15 \cdot 0.15}{0.85 \cdot 0.85 \cdot (-0.85) \cdot 0.15 + 0.15 \cdot 0.15}
\][/tex]
First, we will evaluate the numerator:
1. Calculate [tex]\(0.85 \cdot 0.85 \cdot 0.85\)[/tex]:
[tex]\[
0.85^3 = 0.614125
\][/tex]
2. Calculate [tex]\(0.15 \cdot 0.15 \cdot 0.15\)[/tex]:
[tex]\[
0.15^3 = 0.003375
\][/tex]
3. Add these two results:
[tex]\[
0.614125 + 0.003375 = 0.6175
\][/tex]
So, the numerator is [tex]\(0.6175\)[/tex].
Next, let’s evaluate the denominator:
1. Calculate [tex]\(0.85 \cdot 0.85\)[/tex]:
[tex]\[
0.85^2 = 0.7225
\][/tex]
2. Multiply this by [tex]\(-0.85\)[/tex] and by [tex]\(0.15\)[/tex]:
[tex]\[
0.7225 \cdot (-0.85) \cdot 0.15 = 0.7225 \cdot -0.1275 = -0.09219375
\][/tex]
3. Calculate [tex]\(0.15 \cdot 0.15\)[/tex]:
[tex]\[
0.15^2 = 0.0225
\][/tex]
4. Add these two results:
[tex]\[
-0.09219375 + 0.0225 = -0.06969375
\][/tex]
So, the denominator is approximately [tex]\(-0.06969375\)[/tex].
Finally, we evaluate the fraction by dividing the numerator by the denominator:
[tex]\[
\frac{0.6175}{-0.06969375} \approx -8.869736960229824
\][/tex]
Therefore, the overall result of this expression is [tex]\(\boxed{-8.869736960229824}\)[/tex].
