To solve the expression [tex]\( 0.1 + \sqrt{7} \cdot \frac{1}{8} + \left(\frac{1}{5} + \frac{2}{5}\right) \)[/tex], we'll break it down into smaller parts and evaluate each part step-by-step.
### Step 1: Evaluate [tex]\( 0.1 \)[/tex]
This is the initial number in the expression, and it remains [tex]\( 0.1 \)[/tex].
### Step 2: Evaluate [tex]\( \sqrt{7} \cdot \frac{1}{8} \)[/tex]
First, find the value of [tex]\( \sqrt{7} \)[/tex]:
[tex]\[ \sqrt{7} \approx 2.6457513110645906 \][/tex]
Now, multiply this by [tex]\( \frac{1}{8} \)[/tex]:
[tex]\[ 2.6457513110645906 \cdot \frac{1}{8} \approx 0.33071891388307384 \][/tex]
### Step 3: Evaluate [tex]\( \left(\frac{1}{5} + \frac{2}{5}\right) \)[/tex]
Add the fractions inside the parentheses:
[tex]\[ \frac{1}{5} + \frac{2}{5} = \frac{1+2}{5} = \frac{3}{5} \][/tex]
Converting [tex]\( \frac{3}{5} \)[/tex] to decimal form:
[tex]\[ \frac{3}{5} = 0.6 \][/tex]
### Step 4: Sum all parts
Now, sum all the parts we have calculated:
[tex]\[ 0.1 + 0.33071891388307384 + 0.6 \][/tex]
Adding these together:
[tex]\[ 0.1 + 0.33071891388307384 = 0.43071891388307384 \][/tex]
[tex]\[ 0.43071891388307384 + 0.6 = 1.030718913883074 \][/tex]
### Final Answer
Thus, the value of the expression is:
[tex]\[ 1.030718913883074 \][/tex]
Therefore, the step-by-step evaluated result for the expression [tex]\( 0.1 + \sqrt{7} \cdot \frac{1}{8} + (\frac{1}{5} + \frac{2}{5}) \)[/tex] is [tex]\(\boxed{1.030718913883074}\)[/tex].
