Answer:
[tex]\text{C}.\ \ \dfrac{2(a+b)^2x^4}{y}[/tex]
Step-by-step explanation:
We can simplify the radical expression:
[tex]\sqrt{\dfrac{(a+b)^5x^2y^3}{0.25(a+b)x^{-6}y^5}}[/tex]
using the rules of exponents:
- multiplication: [tex]x^a \cdot x^b = x^{a+b}[/tex]
- division: [tex]x^a / x^b = x^{a-b}[/tex]
First, we can group like terms:
[tex]\sqrt{\dfrac{(a+b)^5}{(a+b)^1}\cdot \dfrac{x^2}{x^{-6}}\cdot \dfrac{y^3}{y^5}\cdot \dfrac{1}{\frac{1}{4}}}[/tex]
Next, we can apply the division rule:
[tex]\sqrt{(a+b)^4\cdot x^{(2-(-6))}\cdot y^{3-5}\cdot 4}[/tex]
Simplifying the exponents, we get:
[tex]\sqrt{(a+b)^4\cdot x^{8}\cdot y^{-2}\cdot 4}[/tex]
Next, we can take the square root of each factor:
[tex](a+b)^2\sqrt{x^{8}\cdot y^{-2}\cdot 4}[/tex]
[tex]=(a+b)^2x^4\sqrt{y^{-2}\cdot 4}[/tex]
[tex]=(a+b)^2x^4y^{-1}\sqrt{ 4}[/tex]
[tex]=2(a+b)^2x^4y^{-1}[/tex]
Finally, we can rewrite the negative exponent with a positive exponent using a fraction:
[tex]2(a+b)^2x^4 \cdot \dfrac{1}{y^1}[/tex]
[tex]=\boxed{\dfrac{2(a+b)^2x^4}{y}}[/tex] (answer option C)
