Certainly! Let's simplify each expression step by step.
### Expression 1: [tex]\(\frac{3a^2 b^{-4}}{12a^{-2}b^{-2}}\)[/tex]
1. Combine the constants:
[tex]\[
\frac{3}{12} = \frac{1}{4}
\][/tex]
2. Combine the powers of [tex]\(a\)[/tex]:
[tex]\[
a^2 \div a^{-2} = a^{2 - (-2)} = a^{2 + 2} = a^4
\][/tex]
3. Combine the powers of [tex]\(b\)[/tex]:
[tex]\[
b^{-4} \div b^{-2} = b^{-4 - (-2)} = b^{-4 + 2} = b^{-2}
\][/tex]
4. Put it all together:
[tex]\[
\frac{3a^2 b^{-4}}{12a^{-2}b^{-2}} = \frac{1}{4} \cdot a^4 \cdot b^{-2} = \frac{a^4}{4b^2}
\][/tex]
Thus, the simplified form is:
[tex]\[
\frac{a^4}{4b^2}
\][/tex]
### Expression 2: [tex]\(\frac{a^4}{4b^2}\)[/tex]
This expression is already in its simplest form:
[tex]\[
\frac{a^4}{4b^2}
\][/tex]
### Expression 3: [tex]\(\frac{1}{4b^6}\)[/tex]
This expression is already in its simplest form:
[tex]\[
\frac{1}{4b^6}
\][/tex]
### Expression 4: [tex]\(\frac{1}{4a^{-4}b^2}\)[/tex]
1. Simplify [tex]\(a^{-4}\)[/tex] in the denominator:
[tex]\[
a^{-4} = \frac{1}{a^4}
\][/tex]
2. Rewrite the expression:
[tex]\[
\frac{1}{4a^{-4}b^2} = \frac{1}{\frac{4b^2}{a^4}} = \frac{a^4}{4b^2}
\][/tex]
Thus, the simplified form is:
[tex]\[
\frac{a^4}{4b^2}
\][/tex]
### Expression 5: [tex]\(\frac{a^4}{4b^6}\)[/tex]
This expression is already in its simplest form:
[tex]\[
\frac{a^4}{4b^6}
\][/tex]
### Summary
The simplified forms of the given expressions are:
1. [tex]\(\frac{3a^2 b^{-4}}{12a^{-2}b^{-2}} = \frac{a^4}{4b^2}\)[/tex]
2. [tex]\(\frac{a^4}{4b^2} = \frac{a^4}{4b^2}\)[/tex]
3. [tex]\(\frac{1}{4b^6} = \frac{1}{4b^6}\)[/tex]
4. [tex]\(\frac{1}{4a^{-4}b^2} = \frac{a^4}{4b^2}\)[/tex]
5. [tex]\(\frac{a^4}{4b^6} = \frac{a^4}{4b^6}\)[/tex]
These are the simplified forms of the given expressions.
