To simplify the expression [tex]\(\left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}\)[/tex], follow these steps:
1. Simplify inside the parenthesis first:
[tex]\[
\frac{20 x^5 y^2}{5 x^{-3} y^7}
\][/tex]
2. Combine the constants:
[tex]\[
\frac{20}{5} = 4
\][/tex]
3. Combine the [tex]\(x\)[/tex]-terms using exponent rules:
[tex]\[
\frac{x^5}{x^{-3}} = x^{5 - (-3)} = x^{5 + 3} = x^8
\][/tex]
4. Combine the [tex]\(y\)[/tex]-terms using exponent rules:
[tex]\[
\frac{y^2}{y^7} = y^{2 - 7} = y^{-5}
\][/tex]
5. Rewrite the simplified expression inside the parenthesis:
[tex]\[
4 x^8 y^{-5}
\][/tex]
6. Apply the [tex]\(-3\)[/tex] exponent to the entire expression:
[tex]\[
\left(4 x^8 y^{-5}\right)^{-3}
\][/tex]
7. Use the power rule [tex]\((a \cdot b \cdot c)^{n} = a^n \cdot b^n \cdot c^n\)[/tex]:
[tex]\[
4^{-3} \cdot (x^8)^{-3} \cdot (y^{-5})^{-3}
\][/tex]
8. Simplify each part separately:
- For [tex]\(4^{-3}\)[/tex]:
[tex]\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\][/tex]
The coefficient term becomes [tex]\(\frac{1}{64}\)[/tex].
- For [tex]\((x^8)^{-3}\)[/tex]:
[tex]\[
(x^8)^{-3} = x^{8 \cdot (-3)} = x^{-24}
\][/tex]
- For [tex]\((y^{-5})^{-3}\)[/tex]:
[tex]\[
(y^{-5})^{-3} = y^{-5 \cdot (-3)} = y^{15}
\][/tex]
9. Combine all parts to obtain the final simplified expression:
[tex]\[
\frac{1}{64} \cdot x^{-24} \cdot y^{15}
\][/tex]
So, the fully simplified form of the expression [tex]\(\left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}\)[/tex] is:
[tex]\[
\frac{1}{64} \cdot x^{-24} \cdot y^{15}
\][/tex]
If you plug in specific numerical values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] (e.g., [tex]\(x = 2\)[/tex], [tex]\(y = 3\)[/tex]), you would get the result:
[tex]\[
(0.015625, 5.960464477539063e-08, 14348907)
\][/tex]
