Sure, let's go through each part step-by-step to simplify the given expressions.
### Part (a)
Simplify [tex]\(\left(x^{-1} y^{-2}\right)^{-3}\)[/tex]:
1. First, recognize the power of a power rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
2. Apply the power of a power rule individually to each component inside the parentheses:
[tex]\[
\left( x^{-1} y^{-2} \right)^{-3} = x^{-1 \cdot -3} \cdot y^{-2 \cdot -3}
\][/tex]
3. Simplify the exponents:
[tex]\[
x^{3} \cdot y^{6}
\][/tex]
So, the simplified form of [tex]\(\left(x^{-1} y^{-2}\right)^{-3}\)[/tex] is [tex]\(x^3 y^6\)[/tex].
### Part (b)
Simplify [tex]\(\left(2 a^{-2} b^2\right)^{-2}\)[/tex]:
1. Again, apply the power of a power rule to each component inside the parentheses:
[tex]\[
\left(2 a^{-2} b^2 \right)^{-2} = 2^{-2} \cdot a^{-2 \cdot -2} \cdot b^{2 \cdot -2}
\][/tex]
2. Simplify each part:
[tex]\[
2^{-2} = \frac{1}{4}, \quad a^{-2 \cdot -2} = a^4, \quad b^{2 \cdot -2} = b^{-4}
\][/tex]
3. Combine the simplified terms:
[tex]\[
\frac{1}{4} \cdot a^4 \cdot b^{-4} = \frac{a^4}{4b^4}
\][/tex]
So, the simplified form of [tex]\(\left(2 a^{-2} b^2\right)^{-2}\)[/tex] is [tex]\(\frac{a^4}{4 b^4}\)[/tex].
### Part (c)
Simplify [tex]\(\left(4 m^2 n^3\right)^{-3}\)[/tex]:
1. Apply the power of a power rule to each component inside the parentheses:
[tex]\[
\left(4 m^2 n^3 \right)^{-3} = 4^{-3} \cdot m^{2 \cdot -3} \cdot n^{3 \cdot -3}
\][/tex]
2. Simplify each part:
[tex]\[
4^{-3} = \frac{1}{64}, \quad m^{2 \cdot -3} = m^{-6}, \quad n^{3 \cdot -3} = n^{-9}
\][/tex]
3. Combine the simplified terms:
[tex]\[
\frac{1}{64} \cdot m^{-6} \cdot n^{-9} = \frac{1}{64 m^6 n^9}
\][/tex]
So, the simplified form of [tex]\(\left(4 m^2 n^3\right)^{-3}\)[/tex] is [tex]\(\frac{1}{64 m^6 n^9}\)[/tex].
### Part (d)
Simplify [tex]\(\left(\frac{3}{2} m^{-2} n^{-3}\right)^{-4}\)[/tex]:
1. Apply the power of a power rule to each component inside the parentheses:
[tex]\[
\left(\frac{3}{2} m^{-2} n^{-3} \right)^{-4} = \left(\frac{3}{2}\right)^{-4} \cdot m^{-2 \cdot -4} \cdot n^{-3 \cdot -4}
\][/tex]
2. Simplify each part:
[tex]\[
\left(\frac{3}{2}\right)^{-4} = \left(\frac{2}{3}\right)^{4} = \frac{16}{81}, \quad m^{-2 \cdot -4} = m^8, \quad n^{-3 \cdot -4} = n^{12}
\][/tex]
3. Combine the simplified terms:
[tex]\[
\frac{16}{81} \cdot m^8 \cdot n^{12} = \frac{16 m^8 n^{12}}{81}
\][/tex]
So, the simplified form of [tex]\(\left(\frac{3}{2} m^{-2} n^{-3}\right)^{-4}\)[/tex] is [tex]\(\frac{16 m^8 n^{12}}{81}\)[/tex].
