To simplify the expression [tex]\(\frac{2}{9x} + \frac{1}{2x^2}\)[/tex], follow these steps:
1. Identify the terms:
- The first term is [tex]\(\frac{2}{9x}\)[/tex].
- The second term is [tex]\(\frac{1}{2x^2}\)[/tex].
2. Find a common denominator for the two fractions:
- The denominators are [tex]\(9x\)[/tex] and [tex]\(2x^2\)[/tex].
- The least common multiple (LCM) of these denominators is [tex]\(18x^2\)[/tex].
3. Rewrite each fraction with the common denominator:
- For the first term [tex]\(\frac{2}{9x}\)[/tex]:
[tex]\[
\frac{2}{9x} \cdot \frac{2x}{2x} = \frac{2 \cdot 2x}{18x^2} = \frac{4x}{18x^2} = \frac{4}{9x}
\][/tex]
- For the second term [tex]\(\frac{1}{2x^2}\)[/tex]:
[tex]\[
\frac{1}{2x^2} \cdot \frac{9}{9} = \frac{1 \cdot 9}{18x^2} = \frac{9}{18x^2} = \frac{1}{2x^2} \cdot \frac{9}{9} = \frac{9}{2x^2}
\][/tex]
4. Combine the fractions:
[tex]\[
\frac{4}{9x} + \frac{9}{2x^2}
\][/tex]
Notice that this already ensures the least common denominator for the combined fraction.
5. Add the fractions:
[tex]\[
\frac{4}{9x} ~\textrm{and}~ \frac{9}{2x^2}
\][/tex]
[tex]\[
\frac{8x^2}{18x^2} + \frac{81}{18x^2} = \frac{8x^2 + 81}{18x^2}
\][/tex]
6. Simplify the result if possible:
The simplified form of [tex]\(\frac{8x^2 + 81}{18x^2}\)[/tex] as a single fraction doesn't reduce further. Therefore, the fully simplified result of the given expression [tex]\(\frac{2}{9x} + \frac{1}{2x^2}\)[/tex] is:
[tex]\[
\frac{8x^2 + 81}{18x^2}
\][/tex]
So, the simplified expression is:
[tex]\[
\boxed{\frac{8x^2 + 81}{18x^2}}
\][/tex]
