Sure, let's solve the given expression step by step:
The problem requires simplifying the following expression:
[tex]\[
\frac{a^2}{b^2} - 3 + \frac{2b^2}{a^2}
\][/tex]
1. Identify the terms in the expression:
- The first term is [tex]\(\frac{a^2}{b^2}\)[/tex].
- The second term is [tex]\(-3\)[/tex].
- The third term is [tex]\(\frac{2b^2}{a^2}\)[/tex].
2. Write the expression with all the terms:
[tex]\[
\frac{a^2}{b^2} - 3 + \frac{2b^2}{a^2}
\][/tex]
3. Combine like terms if possible:
In this case, the terms are not like terms, so they cannot be combined through addition or subtraction.
4. Check for possible factorizations or simplifications:
Since we are dealing with a rational expression, let's see if there's anything common we can factor out.
- The first term [tex]\(\frac{a^2}{b^2}\)[/tex] is already in its simplest form.
- The middle term is a constant, so it stays as [tex]\(-3\)[/tex].
- The third term [tex]\(\frac{2b^2}{a^2}\)[/tex] is also in its simplest form.
5. Rewrite the expression to confirm all parts are considered:
[tex]\[
\frac{a^2}{b^2} - 3 + \frac{2b^2}{a^2}
\][/tex]
Since no further simplification is apparent, the final, simplified expression is:
[tex]\[
\frac{a^2}{b^2} - 3 + \frac{2b^2}{a^2}
\][/tex]
So, the step-by-step simplification results in the same expression.
