Certainly! Let's simplify the expression step by step.
Given:
[tex]\[
\left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}
\][/tex]
1. Simplify inside the parentheses:
- Simplify the coefficients:
[tex]\[
\frac{20}{5} = 4
\][/tex]
- Apply the rules of exponents for the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^5}{x^{-3}} = x^{5 - (-3)} = x^{5 + 3} = x^8
\][/tex]
- Apply the rules of exponents for the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^2}{y^7} = y^{2 - 7} = y^{-5}
\][/tex]
- Combine these simplified pieces:
[tex]\[
\frac{20 x^5 y^2}{5 x^{-3} y^7} = 4 x^8 y^{-5}
\][/tex]
2. Apply the outer exponent of [tex]\(-3\)[/tex]:
- For the coefficient [tex]\(4\)[/tex]:
[tex]\[
4^{-3} = \left(\frac{1}{4}\right)^3 = \frac{1}{4^3} = \frac{1}{64} = 0.015625
\][/tex]
- For the [tex]\(x\)[/tex] term:
[tex]\[
(x^8)^{-3} = x^{8 \cdot (-3)} = x^{-24}
\][/tex]
- For the [tex]\(y\)[/tex] term:
[tex]\[
(y^{-5})^{-3} = y^{-5 \cdot (-3)} = y^{15}
\][/tex]
3. Combine the results:
[tex]\[
\left(4 x^8 y^{-5}\right)^{-3} = 0.015625 x^{-24} y^{15}
\][/tex]
Therefore, the final simplified expression is:
[tex]\[
0.015625 x^{-24} y^{15}
\][/tex]
