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

Step 1: Taking y^(1/2) inside the cube root: We have the expression y^((1)/(2))*root(3)(16x^(2)y^((3)/(2))+8y^(2)). Let's simplify it step by step. First, we can take y^((1)/(2)) inside the cube root as a factor of y, because when we multiply exponents with the same base, we add the exponents.Step 2: Combining y terms inside the cube root: Inside the cube root, we will have y^((1/2) * (3/3)) = y^((3/6)) = y^((1/2)). So, we can combine this with the y terms inside the cube root. The expression inside the cube root becomes 16x^(2)y^((3/2) + (1/2)) + 8y^((2) + (1/2)).Step 3: Simplifying the exponents: Now, we simplify the exponents: y^((3/2) + (1/2)) = y^((4/2)) = y^2 and y^((2) + (1/2)) = y^((5/2)). So, the expression inside the cube root is now 16x^(2)y^2 + 8y^((5/2)).Step 4: Factoring out a common term: We can factor out a 8y^2 from both terms inside the cube root to simplify the expression further. The expression becomes y^((1)/(2))*root(3)(8y^2(2x^(2) + y^((1/2)))).Step 5: Taking the common term outside the cube root: Now, we can take the factor of 8y^2 outside the cube root, remembering that when we take a term outside of a cube root, we divide its exponent by 3. So, 8y^2 becomes 2y^((2/3)) outside the cube root.Step 6: Combining y terms: The expression now is 2y^((2/3))*y^((1)/(2))*root(3)(2x^(2) + y^((1/2))). We need to combine the y terms by adding their exponents: y^((2/3)) * y^((1/2)) = y^((2/3) + (3/6)) = y^((2/3) + (1/2)).Step 7: Final simplification: Simplify the exponents: y^((2/3) + (1/2)) = y^((4/6) + (3/6)) = y^((7/6)). So, the final expression is 2y^((7/6))*root(3)(2x^(2) + y^((1/2))).Step 8: Matching with the answer choices: Looking at the answer choices, we see that this matches with option (A). So, the equivalent expression is 2y^((7/6))*root(3)(2x^(2) + y^((1/2))).

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