To perform the indicated operations and simplify the expression \( \frac{b^2}{6a^2} - \frac{1}{3a^3} + \frac{3b}{8a} \), we'll first find a common denominator for all three terms.
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Identifying the denominators: The denominators we have are \( 6a^2 \), \( 3a^3 \), and \( 8a \).
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Finding the least common multiple (LCM):
- The LCM of the numerical coefficients \( 6, 3, \) and \( 8 \) is \( 24 \).
- The highest power of \( a \) present is \( a^3 \).
- Thus, the LCM of the denominators is \( 24a^3 \).
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Rewriting each fraction with the common denominator:
\[ \frac{b^2}{6a^2} = \frac{b^2 \cdot 4a}{24a^3} = \frac{4ab^2}{24a^3} \]
\[ -\frac{1}{3a^3} = -\frac{1 \cdot 8}{24a^3} = -\frac{8}{24a^3} \]
\[ \frac{3b}{8a} = \frac{3b \cdot 3a^2}{24a^3} = \frac{9ba^2}{24a^3} \]
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Combining the fractions:
Now we can combine the fractions since they have the same denominator:
\[ \frac{4ab^2 - 8 + 9ba^2}{24a^3} \]
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Simplifying:
The expression \( 4ab^2 + 9ba^2 - 8 \) cannot be factored further neatly, so the result remains in this form:
\[ \frac{4ab^2 + 9ba^2 - 8}{24a^3} \]
Now, let's match it with the given options (a, b, c, d).
Here are the operations performed to identify which of these options match the simplified expression:
a) \( b^2 - 1 + \frac{3b}{24a^3} \)
This does not match.
b) \( b^2 + 3b - \frac{1}{11a^6} \)
This does not match.
c) \( b^2 - 1 + \frac{3b}{6a^2 - 3a^3 + 8a} \)
This does not match.
d) \( \frac{4ab^2 - 8 + 9a^2b}{24a^3} \)
This matches our simplified expression.
Final answer:
The simplified form is:
\[ \frac{4ab^2 + 9ba^2 - 8}{24a^3} \]
Thus, the result of performing the indicated operations and simplifying \( \frac{b^2}{6a^2} - \frac{1}{3a^3} + \frac{3b}{8a} \) corresponds with option d:
\[ \frac{4ab^2 - 8 + 9a^2b}{24a^3} \]
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