Given x > 0 and y > 0, select the expression that is equivalent to root(4)(256x^(8)y^(14)) 4x^(32)y^(56) 64x^(2)y^((7)/(2)) 64x^(32)y^(56) 4x^(2)y^((7)/(2))

Identify Given Expression: Identify the given expression and the operation to be performed. We need to find the equivalent expression for the fourth root of 256x^8y^14.Simplify Constant Term: Simplify the constant term under the fourth root. The fourth root of 256 is 4 because 4^4 = 256.Apply Fourth Root to Variables: Apply the fourth root to the variable terms separately. The fourth root of x^8 is x^(8/4) which simplifies to x^2. The fourth root of y^14 is y^(14/4) which simplifies to y^(7/2) because 14/4 = 3.5 or 7/2.Combine Simplified Terms: Combine the simplified terms. The equivalent expression is 4x^2y^(7/2).Check Answer Choices: Check the answer choices to see which one matches the simplified expression. The correct answer choice is 4x^2y^(7/2).

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Precalculus

Operations with rational exponents

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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^{14}}

\newline

4 x^{32} y^{56}

\newline

64 x^{2} y^{\frac{7}{2}}

\newline

64 x^{32} y^{56}

\newline

4 x^{2} y^{\frac{7}{2}}

Identify Given Expression: Identify the given expression and the operation to be performed. $\newline$ We need to find the equivalent expression for the fourth root of $256x^8y^{14}$ .
Simplify Constant Term: Simplify the constant term under the fourth root. $\newline$ The fourth root of $256$ is $4$ because $4^4 = 256$ .
Apply Fourth Root to Variables: Apply the fourth root to the variable terms separately. $\newline$ The fourth root of $x^8$ is $x^{(8/4)}$ which simplifies to $x^2$ . $\newline$ The fourth root of $y^{14}$ is $y^{(14/4)}$ which simplifies to $y^{(7/2)}$ because $14/4 = 3.5$ or $7/2$ .
Combine Simplified Terms: Combine the simplified terms. $\newline$ The equivalent expression is $4x^2y^{(7/2)}$ .
Check Answer Choices: Check the answer choices to see which one matches the simplified expression. $\newline$ The correct answer choice is $4x^2y^{\frac{7}{2}}$ .