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To simplify the expression [tex]\(\frac{2w}{3} - \left\{-\frac{w}{2} + \left[\frac{w}{5} - \frac{2}{5}\right] - \frac{1}{10}\right\} + \left\{\frac{w}{4} - \left[\frac{w}{2} - \frac{1}{2}\right\}\right\} + \frac{w^2}{10} - \frac{2}{5}\)[/tex], let's go through it step-by-step.
1. Simplify inside the curly brackets:
[tex]\[ -\left\{-\frac{w}{2} + \left[\frac{w}{5} - \frac{2}{5}\right] - \frac{1}{10}\right\} \][/tex]
- First, simplify inside the brackets:
[tex]\[ \frac{w}{5} - \frac{2}{5} = \frac{w - 2}{5} \][/tex]
- Then, add the terms:
[tex]\[ -\frac{w}{2} + \frac{w - 2}{5} - \frac{1}{10} \][/tex]
To combine these, find a common denominator (which is 10):
[tex]\[ -\frac{w}{2} = -\frac{5w}{10}, \quad \frac{w - 2}{5} = \frac{2(w - 2)}{10} = \frac{2w - 4}{10}, \quad \frac{1}{10} = \frac{1}{10} \][/tex]
So,
[tex]\[ -\frac{5w}{10} + \frac{2w - 4}{10} - \frac{1}{10} = \frac{-5w + 2w - 4 - 1}{10} = \frac{-3w - 5}{10} \][/tex]
Finally, distribute the negative sign:
[tex]\[ -\left(\frac{-3w - 5}{10}\right) = \frac{3w + 5}{10} \][/tex]
2. Simplify inside the second set of curly brackets:
[tex]\[ \left\{\frac{w}{4} - \left[\frac{w}{2} - \frac{1}{2}\right]\right\} \][/tex]
- First, simplify inside the brackets:
[tex]\[ \frac{w}{2} - \frac{1}{2} \][/tex]
- Combine the terms:
[tex]\[ \frac{w}{4} - \left(\frac{w}{2} - \frac{1}{2}\right) \][/tex]
To combine these, find a common denominator (which is 4):
[tex]\[ \frac{w}{4} = \frac{w}{4}, \quad \frac{w}{2} = \frac{2w}{4}, \quad \frac{1}{2} = \frac{2}{4} \][/tex]
So,
[tex]\[ \frac{w}{4} - \left(\frac{2w}{4} - \frac{2}{4}\right) = \frac{w}{4} - \frac{2w - 2}{4} = \frac{w - 2w + 2}{4} = \frac{-w + 2}{4} = \frac{2 - w}{4} \][/tex]
3. Combine all simplified terms:
[tex]\[ \frac{2w}{3} + \frac{3w + 5}{10} + \frac{2 - w}{4} + \frac{w^2}{10} - \frac{2}{5} \][/tex]
4. Find a common denominator for these terms:
- The common denominator is 60:
[tex]\[ \frac{2w}{3} = \frac{2w \cdot 20}{3 \cdot 20} = \frac{40w}{60}, \quad \frac{3w + 5}{10} = \frac{(3w + 5) \cdot 6}{10 \cdot 6} = \frac{18w + 30}{60}, \quad \frac{2 - w}{4} = \frac{(2 - w) \cdot 15}{4 \cdot 15} = \frac{30 - 15w}{60}, \quad \frac{w^2}{10} = \frac{w^2 \cdot 6}{10 \cdot 6} = \frac{6w^2}{60}, \quad \frac{2}{5} = \frac{2 \cdot 12}{5 \cdot 12} = \frac{24}{60} \][/tex]
Now combine:
[tex]\[ \frac{40w}{60} + \frac{18w + 30}{60} + \frac{30 - 15w}{60} + \frac{6w^2}{60} - \frac{24}{60} = \frac{6w^2 + 40w + 18w + 30 + 30 - 15w - 24}{60} = \frac{6w^2 + 43w + 36}{60} \][/tex]
5. Final simplified expression:
After simplifying the numerator and ensuring all forms are correct:
[tex]\[ 0.1w^2 + 0.716666666666667w + 0.6 \][/tex]
Therefore, the simplified expression is [tex]\(\boxed{0.1w^2 + 0.716666666666667w + 0.6}\)[/tex].
1. Simplify inside the curly brackets:
[tex]\[ -\left\{-\frac{w}{2} + \left[\frac{w}{5} - \frac{2}{5}\right] - \frac{1}{10}\right\} \][/tex]
- First, simplify inside the brackets:
[tex]\[ \frac{w}{5} - \frac{2}{5} = \frac{w - 2}{5} \][/tex]
- Then, add the terms:
[tex]\[ -\frac{w}{2} + \frac{w - 2}{5} - \frac{1}{10} \][/tex]
To combine these, find a common denominator (which is 10):
[tex]\[ -\frac{w}{2} = -\frac{5w}{10}, \quad \frac{w - 2}{5} = \frac{2(w - 2)}{10} = \frac{2w - 4}{10}, \quad \frac{1}{10} = \frac{1}{10} \][/tex]
So,
[tex]\[ -\frac{5w}{10} + \frac{2w - 4}{10} - \frac{1}{10} = \frac{-5w + 2w - 4 - 1}{10} = \frac{-3w - 5}{10} \][/tex]
Finally, distribute the negative sign:
[tex]\[ -\left(\frac{-3w - 5}{10}\right) = \frac{3w + 5}{10} \][/tex]
2. Simplify inside the second set of curly brackets:
[tex]\[ \left\{\frac{w}{4} - \left[\frac{w}{2} - \frac{1}{2}\right]\right\} \][/tex]
- First, simplify inside the brackets:
[tex]\[ \frac{w}{2} - \frac{1}{2} \][/tex]
- Combine the terms:
[tex]\[ \frac{w}{4} - \left(\frac{w}{2} - \frac{1}{2}\right) \][/tex]
To combine these, find a common denominator (which is 4):
[tex]\[ \frac{w}{4} = \frac{w}{4}, \quad \frac{w}{2} = \frac{2w}{4}, \quad \frac{1}{2} = \frac{2}{4} \][/tex]
So,
[tex]\[ \frac{w}{4} - \left(\frac{2w}{4} - \frac{2}{4}\right) = \frac{w}{4} - \frac{2w - 2}{4} = \frac{w - 2w + 2}{4} = \frac{-w + 2}{4} = \frac{2 - w}{4} \][/tex]
3. Combine all simplified terms:
[tex]\[ \frac{2w}{3} + \frac{3w + 5}{10} + \frac{2 - w}{4} + \frac{w^2}{10} - \frac{2}{5} \][/tex]
4. Find a common denominator for these terms:
- The common denominator is 60:
[tex]\[ \frac{2w}{3} = \frac{2w \cdot 20}{3 \cdot 20} = \frac{40w}{60}, \quad \frac{3w + 5}{10} = \frac{(3w + 5) \cdot 6}{10 \cdot 6} = \frac{18w + 30}{60}, \quad \frac{2 - w}{4} = \frac{(2 - w) \cdot 15}{4 \cdot 15} = \frac{30 - 15w}{60}, \quad \frac{w^2}{10} = \frac{w^2 \cdot 6}{10 \cdot 6} = \frac{6w^2}{60}, \quad \frac{2}{5} = \frac{2 \cdot 12}{5 \cdot 12} = \frac{24}{60} \][/tex]
Now combine:
[tex]\[ \frac{40w}{60} + \frac{18w + 30}{60} + \frac{30 - 15w}{60} + \frac{6w^2}{60} - \frac{24}{60} = \frac{6w^2 + 40w + 18w + 30 + 30 - 15w - 24}{60} = \frac{6w^2 + 43w + 36}{60} \][/tex]
5. Final simplified expression:
After simplifying the numerator and ensuring all forms are correct:
[tex]\[ 0.1w^2 + 0.716666666666667w + 0.6 \][/tex]
Therefore, the simplified expression is [tex]\(\boxed{0.1w^2 + 0.716666666666667w + 0.6}\)[/tex].
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