$y^{\frac{1}{2}} \cdot \sqrt[3]{16 x^{2} y^{\frac{3}{2}}+8 y^{2}}$ $\newline$ Which of the following expressions is equivalent to the given expression assuming $y \geq 0$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $2 y \sqrt[3]{2 x^{2}+y^{\frac{1}{2}}}$ $\newline$ (B) $\sqrt[3]{24 x^{2} y^{\frac{7}{2}}} \cdot y^{\frac{1}{2}}$ $\newline$ (C) $y \sqrt[3]{16 x^{2}}+2 y^{\frac{7}{6}}$ $\newline$ (D) $\sqrt[3]{16 x^{2} y^{2}+8 y^{\frac{5}{2}}}$

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Algebra 1

Compare linear and exponential growth

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y^{\frac{1}{2}} \cdot \sqrt[3]{16 x^{2} y^{\frac{3}{2}}+8 y^{2}}

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Which of the following expressions is equivalent to the given expression assuming

y \geq 0

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Choose

1

answer:

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(A)

2 y \sqrt[3]{2 x^{2}+y^{\frac{1}{2}}}

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(B)

\sqrt[3]{24 x^{2} y^{\frac{7}{2}}} \cdot y^{\frac{1}{2}}

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(C)

y \sqrt[3]{16 x^{2}}+2 y^{\frac{7}{6}}

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(D)

\sqrt[3]{16 x^{2} y^{2}+8 y^{\frac{5}{2}}}

Step $1$ : Taking $y^{(1/2)}$ inside the cube root: We have the expression $y^{(1/2)}\sqrt[3]{16x^{2}y^{(3/2)}+8y^{2}}$ . Let's simplify it step by step. $\newline$ 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 $y^{(\frac{1}{2} \cdot \frac{3}{3})} = y^{(\frac{3}{6})} = y^{(\frac{1}{2})}$ . So, we can combine this with the y terms inside the cube root. $\newline$ The expression inside the cube root becomes $16x^{2}y^{(\frac{3}{2} + \frac{1}{2})} + 8y^{(2 + \frac{1}{2})}$ .
Step $3$ : Simplifying the exponents: Now, we simplify the exponents: $y^{(\frac{3}{2}) + (\frac{1}{2})} = y^{(\frac{4}{2})} = y^2$ and $y^{(2) + (\frac{1}{2})} = y^{(\frac{5}{2})}$ . So, the expression inside the cube root is now $16x^{2}y^2 + 8y^{(\frac{5}{2})}$ .
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. $\newline$ The expression becomes $y^{\left(\frac{1}{2}\right)}\sqrt[3]{8y^2(2x^{2} + y^{\left(\frac{1}{2}\right)})}$ .
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 $3$ . $\newline$ So, $8y^2$ becomes $2y^{(2/3)}$ outside the cube root.
Step $6$ : Combining y terms: The expression now is $2y^{(2/3)} \cdot y^{(1/2)} \cdot \sqrt[3]{2x^2 + y^{(1/2)}}$ . $\newline$ We need to combine the y terms by adding their exponents: $y^{(2/3)} \cdot y^{(1/2)} = y^{(2/3) + (3/6)} = y^{(2/3) + (1/2)}$ .
Step $7$ : Final simplification: Simplify the exponents: $y^{\left(\frac{2}{3}\right) + \left(\frac{1}{2}\right)} = y^{\left(\frac{4}{6}\right) + \left(\frac{3}{6}\right)} = y^{\left(\frac{7}{6}\right)}$ . $\newline$ So, the final expression is $2y^{\left(\frac{7}{6}\right)}\sqrt[3]{2x^{2} + y^{\left(\frac{1}{2}\right)}}$ .
Step $8$ : Matching with the answer choices: Looking at the answer choices, we see that this matches with option (A). $\newline$ So, the equivalent expression is $2y^{(\frac{7}{6})}\sqrt[3]{2x^{2} + y^{(\frac{1}{2})}}$ .