Given x > 0 and y > 0, select the expression that is equivalent to root(4)(256x^(8)y^(10)) 64x^(4)y^(6) 4x^(2)y^((5)/(2)) 4x^(4)y^(6) 64x^(2)y^((5)/(2))

Identify Given Expression: Identify the given expression and the operation to be performed. The given expression is the fourth root of 256x^8y^10, which can be written as (256x^8y^10)^(1/4).Simplify Constant Term: Simplify the constant term under the fourth root. The fourth root of 256 is 4 because 4^4 = 256. So, (256)^(1/4) = 4.Simplify Variable Terms: Simplify the variable terms under the fourth root. For x^8, the fourth root is x^(8/4) = x^2. For y^10, the fourth root is y^(10/4) = y^(5/2) or y^2.5.Combine Simplified Terms: Combine the simplified terms. The expression equivalent to the fourth root of 256x^8y^10 is 4x^2y^(5/2).

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Q. Given

x>0

and

y>0

, select the expression that is equivalent to

\newline

\sqrt[4]{256 x^{8} y^{10}}

\newline

64 x^{4} y^{6}

\newline

4 x^{2} y^{\frac{5}{2}}

\newline

4 x^{4} y^{6}

\newline

64 x^{2} y^{\frac{5}{2}}

Identify Given Expression: Identify the given expression and the operation to be performed. $\newline$ The given expression is the fourth root of $256x^8y^{10}$ , which can be written as $(256x^8y^{10})^{\frac{1}{4}}$ .
Simplify Constant Term: Simplify the constant term under the fourth root. $\newline$ The fourth root of $256$ is $4$ because $4^4 = 256$ . $\newline$ So, $(256)^{1/4} = 4$ .
Simplify Variable Terms: Simplify the variable terms under the fourth root. $\newline$ For $x^8$ , the fourth root is $x^{(8/4)} = x^2$ . $\newline$ For $y^{10}$ , the fourth root is $y^{(10/4)} = y^{(5/2)}$ or $y^{2.5}$ .
Combine Simplified Terms: Combine the simplified terms. The expression equivalent to the fourth root of $256x^8y^{10}$ is $4x^2y^{\frac{5}{2}}$ .