Answer:
[tex]\dfrac{16y^4}{x^{8}}[/tex]
Step-by-step explanation:
To simplify the expression [tex]\sf (\dfrac{2x^5}{8xy^2})^{-2}[/tex], we'll follow these steps:
Apply the Negative Exponent Rule:
[tex]\large\boxed{\boxed{\sf \left(\dfrac{a}{b}\right)^{-n} = \dfrac{1}{(\dfrac{a}{b})^n} = \dfrac{b^n}{a^n}}}[/tex]
Therefore,
[tex]\sf\left(\dfrac{2x^5}{8xy^2}\right)^{-2} = \left(\dfrac{8xy^2}{2x^5}\right)^{2}[/tex]
Simplify Inside the Parentheses:
Simplify [tex]\sf \dfrac{8xy^2}{2x^5}[/tex]:
[tex]\sf\dfrac{8xy^2}{2x^5} = \dfrac{8}{2} \cdot \dfrac{y^2}{x^5-1} = 4 \cdot \dfrac{y^2}{x^4}[/tex]
Apply Exponent Rules:
[tex]\large\boxed{\boxed{(x^a y^b)^c = x^{ac}y^{bc}}}[/tex]
Now, raise the simplified expression to the power of 2:
[tex]\sf \left(4 \cdot \dfrac{y^2}{x^5}\right)^{2} = 4^2 \cdot \left(\dfrac{y^2}{x^4}\right)^2[/tex]
Perform the Calculations:
Calculate [tex]\sf 4^2 = 16[/tex], and square the terms inside the parentheses:
[tex]\sf16 \cdot \left(\dfrac{y^2}{x^5}\right)^2 = 16 \cdot \dfrac{y^{2 \cdot 2 }}{x^{4\cdot 2}} \\\\ = 16 \cdot \dfrac{y^{8}}{x^{8}}[/tex]
Final Simplification:
Combine the terms to get the final simplified expression:
[tex]\sf 16 \cdot \dfrac{y^4}{x^{8}} = \dfrac{16y^4}{x^{8}}[/tex]
Therefore, [tex]\sf \left(\dfrac{2x^5}{8xy^2}\right)^{-2}[/tex] simplifies to [tex]\sf \dfrac{16y^4}{x^{8}}[/tex].
The final simplified expression is:
[tex]\sf \boxed{\dfrac{16y^4}{x^{8}}}[/tex]
