Perform the indicated operations, then simplify.

[tex]\[ \frac{1}{6 a^2}-\frac{5 b}{3 a^3}+\frac{b^2}{4 a} \][/tex]

A. [tex]\(\frac{b^2-5 b+1}{12 a^3}\)[/tex]

B. [tex]\(\frac{3 a^2 b^2+2 a-20 b}{12 a^3}\)[/tex]

C. [tex]\(\frac{1-5 b+b^2}{6 a^2-3 a^3+4 a}\)[/tex]

D. [tex]\(\frac{b^2-5 b+1}{7 a^6}\)[/tex]



Answer :

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]