To simplify the expression \(-\frac{2}{3} \left(\frac{1}{3}g - \frac{3}{4}\right) - 10g\), we'll start by distributing \(-\frac{2}{3}\) into the parentheses.
- Distribute \(-\frac{2}{3}\):
\[ -\frac{2}{3} \cdot \frac{1}{3}g - \left(-\frac{2}{3} \cdot \frac{3}{4}\right) \]
Calculating each term individually:
- \(-\frac{2}{3} \cdot \frac{1}{3}g = -\frac{2}{9}g\)
- \(-\frac{2}{3} \cdot -\frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12} = \frac{1}{2}\)
So we have:
\[ -\frac{2}{9}g + \frac{1}{2} \]
- Now combine this result with \(-10g\):
\[ -\frac{2}{9}g + \frac{1}{2} - 10g \]
To combine the \(g\) terms:
\(-10g\) can be represented as \(-\frac{90}{9}g\) (to get a common denominator of 9).
Thus:
\[ -\frac{2}{9}g - \frac{90}{9}g = -\frac{2 + 90}{9}g = -\frac{92}{9}g \]
- Therefore, the expression now is:
\[ -\frac{92}{9}g + \frac{1}{2} \]
This form is convenient. To express the entire expression with a common denominator, we can convert \(\frac{1}{2}\) to have a denominator of 18:
\[ \frac{1}{2} = \frac{9}{18} \]
Now, we convert \(-\frac{92}{9}g\) to have a denominator of 18, which is:
\[ -\frac{92}{9}g = -\frac{184}{18}g \]
Now, we can write the entire expression as:
\[ -\frac{184}{18}g + \frac{9}{18} \]
Combining these into a single fraction:
\[ -\frac{184g - 9}{18} \]
So the simplified expression is:
\[ -\frac{184}{18}g + \frac{9}{18} \]
While it can be delivered in individual parts, traditionally we combine like terms where they apply. In conclusion, the final simplified expression is:
\[ -\frac{92}{9}g + \frac{1}{2} \quad \text{or} \quad -\frac{184g - 9}{18} \]
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