Let's solve the expression step by step:
Given expression:
[tex]\[
\left(-2 \frac{1}{5} + 0.3\right) + \sqrt[3]{\frac{8}{27}} - \frac{1^5}{\sqrt{25}}
\][/tex]
### Step 1: Simplify the Mixed Number Term
First, convert the mixed number [tex]\( -2 \frac{1}{5} \)[/tex] to an improper fraction or a decimal:
[tex]\[
-2 \frac{1}{5} = -2 - \frac{1}{5}
\][/tex]
[tex]\[
\frac{1}{5} = 0.2
\][/tex]
[tex]\[
-2 - 0.2 = -2.2
\][/tex]
### Step 2: Add 0.3 to the First Term
Next, add 0.3 to [tex]\(-2.2\)[/tex]:
[tex]\[
-2.2 + 0.3 = -1.9
\][/tex]
So, the first part of the expression is:
[tex]\[
\left(-2 \frac{1}{5} + 0.3\right) = -1.9
\][/tex]
### Step 3: Calculate the Cube Root Term
Calculate the cube root of [tex]\(\frac{8}{27}\)[/tex]:
[tex]\[
\frac{8}{27} = \left(\frac{2}{3}\right)^3
\][/tex]
[tex]\[
\sqrt[3]{\frac{8}{27}} = \frac{2}{3}
\][/tex]
### Step 4: Simplify the Fraction Term with Exponents
Calculate [tex]\(\frac{1^5}{\sqrt{25}}\)[/tex]:
[tex]\[
1^5 = 1
\][/tex]
[tex]\[
\sqrt{25} = 5
\][/tex]
[tex]\[
\frac{1^5}{\sqrt{25}} = \frac{1}{5} = 0.2
\][/tex]
### Step 5: Sum All the Terms
Now we can put all the simplified terms together:
[tex]\[
-1.9 + \sqrt[3]{\frac{8}{27}} - \frac{1^5}{\sqrt{25}}
\][/tex]
Substitute the simplified terms:
[tex]\[
-1.9 + \frac{2}{3} - 0.2
\][/tex]
[tex]\[
-1.9 + 0.6666666666666666 - 0.2
\][/tex]
### Step 6: Perform the Final Calculation
Finally, combine the terms:
[tex]\[
-1.9 + 0.6666666666666666 - 0.2 = -1.4333333333333333
\][/tex]
### Summary of Results:
[tex]\[
\left(-2 \frac{1}{5} + 0.3\right) = -1.9
\][/tex]
[tex]\[
\sqrt[3]{\frac{8}{27}} = 0.6666666666666666
\][/tex]
[tex]\[
\frac{1^5}{\sqrt{25}} = 0.2
\][/tex]
So, the value of the original expression is:
[tex]\[
-1.4333333333333333
\][/tex]
