Sure, let's simplify each expression step-by-step.
### Expression 1: [tex]\(\left(\frac{a^2 b^{-3}}{a^{-2} b^2}\right)^2\)[/tex]
1. Simplify Inside the Parentheses:
The expression inside the parentheses is [tex]\(\frac{a^2 b^{-3}}{a^{-2} b^2}\)[/tex].
- For the [tex]\(a\)[/tex] terms:
[tex]\[
\frac{a^2}{a^{-2}} = a^{2 - (-2)} = a^{2 + 2} = a^4
\][/tex]
- For the [tex]\(b\)[/tex] terms:
[tex]\[
\frac{b^{-3}}{b^2} = b^{-3 - 2} = b^{-3 - 2} = b^{-5}
\][/tex]
The simplified expression inside the parentheses is [tex]\(a^4 b^{-5}\)[/tex].
2. Square the Result:
Now we square [tex]\(a^4 b^{-5}\)[/tex]:
[tex]\[
\left(a^4 b^{-5}\right)^2 = a^{4 \times 2} \cdot b^{-5 \times 2} = a^8 \cdot b^{-10} = \frac{a^8}{b^{10}}
\][/tex]
So this simplifies to:
[tex]\[
\frac{a^8}{b^{10}}
\][/tex]
### Expression 2: [tex]\(\frac{a^4}{b^5}\)[/tex]
This expression is already simplified and remains:
[tex]\[
\frac{a^4}{b^5}
\][/tex]
### Expression 3: [tex]\(\frac{a^6}{b^7}\)[/tex]
This expression is also already simplified and remains:
[tex]\[
\frac{a^6}{b^7}
\][/tex]
### Expression 4: [tex]\(\frac{a^8}{b^{10}}\)[/tex]
This expression is already simplified and remains:
[tex]\[
\frac{a^8}{b^{10}}
\][/tex]
### Expression 5: [tex]\(\frac{1}{a^8 b^{10}}\)[/tex]
This can be written as:
[tex]\[
\frac{1}{a^8 b^{10}} = \frac{1}{a^8} \cdot \frac{1}{b^{10}}
\][/tex]
Thus, this remains as:
[tex]\[
\frac{1}{a^8 b^{10}}
\][/tex]
So, the simplified expressions are:
1. [tex]\(\frac{a^8}{b^{10}}\)[/tex]
2. [tex]\(\frac{a^4}{b^5}\)[/tex]
3. [tex]\(\frac{a^6}{b^7}\)[/tex]
4. [tex]\(\frac{a^8}{b^{10}}\)[/tex]
5. [tex]\(\frac{1}{a^8 b^{10}}\)[/tex]
Each expression is now in its simplest form.
