y^((1)/(2))*root(3)(16x^(2)y^((3)/(2))+8y^(2)) Which of the following expressions is equivalent to the given expression assuming y = 0 ? Choose 1 answer: (A) 2yroot(3)(2x^(2)+y^((1)/(2))) (B) root(3)(24x^(2)y^((7)/(2)))*y^((1)/(2)) (c) yroot(3)(16x^(2))+2y^((7)/(6)) (D)root(3)(16x^(2)y^(2)+8y^((5)/(2)))

Question

y^((1)/(2))*root(3)(16x^(2)y^((3)/(2))+8y^(2)) Which of the following expressions is equivalent to the given expression assuming  y >= 0 ? Choose 1 answer: (A)  2yroot(3)(2x^(2)+y^((1)/(2))) (B)  root(3)(24x^(2)y^((7)/(2)))*y^((1)/(2)) (c)  yroot(3)(16x^(2))+2y^((7)/(6))  (D)root(3)(16x^(2)y^(2)+8y^((5)/(2)))

Bytelearn · Accepted Answer

Factor out common term: Factor out the common term from the cubic root. We notice that both terms inside the cubic root have a common factor of 8y^((3)/(2)). We can factor this out to simplify the expression.Factor expression inside root: Factor the expression inside the cubic root. 16x^(2)y^((3)/(2)) + 8y^((2)) = 8y^((3)/(2))(2x^(2) + y^((1)/(2))) Now we have the expression y^((1)/(2))*root(3)(8y^((3)/(2))(2x^(2) + y^((1)/(2)))).Separate cubic root of product: Apply the property of roots to separate the cubic root of the product. root(3)(8y^((3)/(2))(2x^(2) + y^((1)/(2)))) = root(3)(8y^((3)/(2))) * root(3)(2x^(2) + y^((1)/(2)))Simplify cubic root: Simplify the cubic root of 8y^((3)/(2)). Since 8 is 2^3 and y^((3)/(2)) is the cube of y^((1)/(2)), the cubic root of 8y^((3)/(2)) is 2y^((1)/(2)).Combine with remaining part: Combine the simplified cubic root with the remaining part of the expression. We now have y^((1)/(2)) * (2y^((1)/(2)) * root(3)(2x^(2) + y^((1)/(2)))).Multiply terms: Multiply y^((1)/(2)) by 2y^((1)/(2)). Multiplying these two terms gives us 2y^((1)/(2) + (1)/(2)) = 2y^((2)/(2)) = 2y.Combine with cubic root: Combine the result with the remaining cubic root. The final expression is 2y * root(3)(2x^(2) + y^((1)/(2))).

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