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Given 
x > 0 and 
y > 0, select the expression that is equivalent to

root(4)(256x^(2)y^(8))

64x^(2)y^((1)/(2))

4x^(2)y^((1)/(2))

64x^((1)/(2))y^(2)

4x^((1)/(2))y^(2)

Given x>0 x>0 and y>0 y>0 , select the expression that is equivalent to\newline256x2y84 \sqrt[4]{256 x^{2} y^{8}} \newline64x2y12 64 x^{2} y^{\frac{1}{2}} \newline4x2y12 4 x^{2} y^{\frac{1}{2}} \newline64x12y2 64 x^{\frac{1}{2}} y^{2} \newline4x12y2 4 x^{\frac{1}{2}} y^{2}

Full solution

Q. Given x>0 x>0 and y>0 y>0 , select the expression that is equivalent to\newline256x2y84 \sqrt[4]{256 x^{2} y^{8}} \newline64x2y12 64 x^{2} y^{\frac{1}{2}} \newline4x2y12 4 x^{2} y^{\frac{1}{2}} \newline64x12y2 64 x^{\frac{1}{2}} y^{2} \newline4x12y2 4 x^{\frac{1}{2}} y^{2}
  1. Write Expression with Radical Notation: Write the given expression with radical notation.\newlineThe fourth root of 256x2y8256x^2y^8 can be written as (256x2y8)14(256x^2y^8)^{\frac{1}{4}}.
  2. Simplify Constant Term: Simplify the constant term under the fourth root.\newline256256 is a perfect fourth power since 256=44256 = 4^4. Therefore, the fourth root of 256256 is 44.\newline(256x2y8)1/4=(44x2y8)1/4(256x^2y^8)^{1/4} = (4^4x^2y^8)^{1/4}
  3. Apply Exponent Property: Apply the property of exponents (am)n=amn(a^m)^n = a^{m*n} to simplify the expression.\newline(44x2y8)14=44(14)x2(14)y8(14)(4^{4}x^{2}y^{8})^{\frac{1}{4}} = 4^{4*(\frac{1}{4})} * x^{2*(\frac{1}{4})} * y^{8*(\frac{1}{4})}
  4. Simplify Exponents: Simplify the exponents by multiplying them.\newline44×(14)=41=44^{4\times(\frac{1}{4})} = 4^1 = 4\newlinex2×(14)=x12x^{2\times(\frac{1}{4})} = x^{\frac{1}{2}}\newliney8×(14)=y2y^{8\times(\frac{1}{4})} = y^2
  5. Combine Simplified Terms: Combine the simplified terms. 4×x12×y24 \times x^{\frac{1}{2}} \times y^2
  6. Compare with Given Options: Compare the result with the given options.\newlineThe expression 4×x12×y24 \times x^{\frac{1}{2}} \times y^2 matches the option "4x(12)y24x^{\left(\frac{1}{2}\right)}y^{2}".

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