Given x 0 and y 0, select the expression that is equivalent to root(4)(81x^(2)y^(18)) 9x^(8)y^(72) 9x^((1)/(2))y^((9)/(2)) 3x^(8)y^(72) 3x^((1)/(2))y^((9)/(2))

Question

Given  x > 0 and  y > 0, select the expression that is equivalent to  root(4)(81x^(2)y^(18))  9x^(8)y^(72)  9x^((1)/(2))y^((9)/(2))  3x^(8)y^(72)  3x^((1)/(2))y^((9)/(2))

Bytelearn · Accepted Answer

Identify Given Expression: Identify the given expression and the operation to be performed. The given expression is the fourth root of 81x^(2)y^(18), which can be written as (81x^(2)y^(18))^(1/4).Break Down Prime Factors: Break down the expression inside the root into prime factors and powers that are easily manageable with the fourth root. 81 is a perfect square and can be written as 3^4. The variables x and y are already in power form. (81x^(2)y^(18))^(1/4) = (3^4x^(2)y^(18))^(1/4)Apply Exponent Property: Apply the property of exponents which states that (a^m)^n = a^(m*n) to simplify the expression. (3^4x^(2)y^(18))^(1/4) = 3^(4*(1/4))x^(2*(1/4))y^(18*(1/4))Simplify Exponents: Simplify the exponents by multiplying them. 3^(4*(1/4)) = 3^1 = 3 x^(2*(1/4)) = x^(1/2) y^(18*(1/4)) = y^(18/4) = y^(9/2)Combine Simplified Terms: Combine the simplified terms to get the final expression. The equivalent expression is 3x^(1/2)y^(9/2).

Given x>0 and y>0 , select the expression that is equivalent to $\newline$ $\sqrt[4]{81 x^{2} y^{18}}$ $\newline$ $9 x^{8} y^{72}$ $\newline$ $9 x^{\frac{1}{2}} y^{\frac{9}{2}}$ $\newline$ $3 x^{8} y^{72}$ $\newline$ $3 x^{\frac{1}{2}} y^{\frac{9}{2}}$

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