Let's solve each part step-by-step starting with the expression [tex]\(\frac{1}{6a^2} - \frac{5b}{3a^3} + \frac{b^2}{4a}\)[/tex].
Step 1: Simplify the given expression [tex]\(\frac{1}{6a^2} - \frac{5b}{3a^3} + \frac{b^2}{4a}\)[/tex]
Start by finding a common denominator. The denominators are [tex]\(6a^2\)[/tex], [tex]\(3a^3\)[/tex], and [tex]\(4a\)[/tex]. The least common multiple (LCM) of these denominators is [tex]\(12a^3\)[/tex].
Rewrite each fraction with the common denominator [tex]\(12a^3\)[/tex]:
[tex]\[
\frac{1}{6a^2} = \frac{1 \cdot 2a}{6a^2 \cdot 2a} = \frac{2a}{12a^3}
\][/tex]
[tex]\[
\frac{5b}{3a^3} = \frac{5b \cdot 4}{3a^3 \cdot 4} = \frac{20b}{12a^3}
\][/tex]
[tex]\[
\frac{b^2}{4a} = \frac{b^2 \cdot 3a^2}{4a \cdot 3a^2} = \frac{3a^2b^2}{12a^3}
\][/tex]
Now, combine these fractions:
[tex]\[
\frac{2a}{12a^3} - \frac{20b}{12a^3} + \frac{3a^2b^2}{12a^3}
\][/tex]
Combine them into a single fraction:
[tex]\[
\frac{2a - 20b + 3a^2b^2}{12a^3}
\][/tex]
This expression cannot be simplified further, so we keep it as:
[tex]\[
\frac{2a - 20b + 3a^2b^2}{12a^3}
\][/tex]
Step 2: Simplify the given fraction [tex]\(\frac{b^2 - 5b + 1}{12a^3}\)[/tex]
This fraction is already in its simplest form since the numerator and denominator share no common factors:
[tex]\[
\frac{b^2 - 5b + 1}{12a^3}
\][/tex]
Step 3: Simplify the given fraction [tex]\(\frac{3a^2 b^2 + 2a - 20b}{12a^3}\)[/tex]
This fraction is also in its simplest form, as the numerator and denominator share no common factors:
[tex]\[
\frac{3a^2 b^2 + 2a - 20b}{12a^3}
\][/tex]
Step 4: Simplify the given fraction [tex]\(\frac{1 - 5b + b^2}{6a^2 - 3a^3 + 4a}\)[/tex]
This fraction is already in its simplest form, as the numerator and denominator share no common factors:
[tex]\[
\frac{1 - 5b + b^2}{6a^2 - 3a^3 + 4a}
\][/tex]
Step 5: Simplify the given fraction [tex]\(\frac{b^2 - 5b + 1}{7a^6}\)[/tex]
This fraction is already in its simplest form, as the numerator and denominator share no common factors:
[tex]\[
\frac{b^2 - 5b + 1}{7a^6}
\][/tex]
Final Result:
[tex]\[
\text{Simplified forms:} \\
\frac{2a - 20b + 3a^2b^2}{12a^3}, \quad \frac{b^2 - 5b + 1}{12a^3}, \quad \frac{3a^2 b^2 + 2a - 20b}{12a^3}, \quad \frac{1 - 5b + b^2}{6a^2 - 3a^3 + 4a}, \quad \frac{b^2 - 5b + 1}{7a^6}
\][/tex]
