Sure, let's solve the given expression step by step.
We need to evaluate the following expression:
[tex]\[
\frac{2.5^3 - 1.4^3}{(2.5)^2 + 2.5 \times 1.4 + (1.4)^2}
\][/tex]
### Step 1: Calculate the numerator
The numerator of the fraction is [tex]\(2.5^3 - 1.4^3\)[/tex].
First, let's calculate [tex]\(2.5^3\)[/tex]:
[tex]\[
2.5^3 = 2.5 \times 2.5 \times 2.5 = 15.625
\][/tex]
Next, let's calculate [tex]\(1.4^3\)[/tex]:
[tex]\[
1.4^3 = 1.4 \times 1.4 \times 1.4 = 2.744
\][/tex]
Now, subtract the two results:
[tex]\[
15.625 - 2.744 = 12.881
\][/tex]
So, the numerator is 12.881.
### Step 2: Calculate the denominator
The denominator of the fraction is [tex]\((2.5)^2 + 2.5 \times 1.4 + (1.4)^2\)[/tex].
First, let's calculate [tex]\((2.5)^2\)[/tex]:
[tex]\[
(2.5)^2 = 2.5 \times 2.5 = 6.25
\][/tex]
Next, calculate [tex]\(2.5 \times 1.4\)[/tex]:
[tex]\[
2.5 \times 1.4 = 3.5
\][/tex]
Then, calculate [tex]\((1.4)^2\)[/tex]:
[tex]\[
(1.4)^2 = 1.4 \times 1.4 = 1.96
\][/tex]
Now, sum these three results:
[tex]\[
6.25 + 3.5 + 1.96 = 11.71
\][/tex]
So, the denominator is 11.71.
### Step 3: Divide the numerator by the denominator
[tex]\[
\frac{12.881}{11.71} = 1.1
\][/tex]
So, the final result of the given expression is:
[tex]\[
1.1
\][/tex]
Thus, we have:
[tex]\[
\frac{2.5^3 - 1.4^3}{(2.5)^2 + 2.5 \times 1.4 + (1.4)^2} = 1.1
\][/tex]
The result is [tex]\(\boxed{1.1}\)[/tex].
