$4 / 4$ $\newline$ $75582786$ $\newline$ Which radical expression is equivalent to $y^{\frac{3}{8}}$ ? $\newline$ $\frac{\sqrt[3]{y}}{\sqrt[8]{y}} \quad(\sqrt[3]{y})^{8}$ $\newline$ Anthony $\newline$ Bautista

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Algebra 2

Simplify radical expressions with root inside the root

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4 / 4

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75582786

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Which radical expression is equivalent to

y^{\frac{3}{8}}

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\frac{\sqrt[3]{y}}{\sqrt[8]{y}} \quad(\sqrt[3]{y})^{8}

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Anthony

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Bautista

Understand the Problem: First, let's understand the problem. We need to find a radical expression that is equivalent to $y$ raised to the power of $\frac{3}{8}$ .
Consider Given Options: Now, let's consider the given options and see which one is equivalent to $y^{\frac{3}{8}}$ .
Option $1$ : $\frac{\sqrt[3]{y}}{\sqrt[8]{y}}$
Option $2$ : $(\sqrt[3]{y})^8$
We need to express these options in exponential form to compare them with $y^{\frac{3}{8}}$ .
Convert Option $1$ : Let's convert Option $1$ to exponential form. $\newline$ $\frac{\sqrt[3]{y}}{\sqrt[8]{y}} = y^{\frac{1}{3}} / y^{\frac{1}{8}}$ $\newline$ Using the quotient rule of exponents, we subtract the exponents. $\newline$ $y^{\frac{1}{3} - \frac{1}{8}} = y^{\frac{8}{24} - \frac{3}{24}} = y^{\frac{5}{24}}$ $\newline$ This is not equivalent to $y^{\frac{3}{8}}$ .
Convert Option $2$ : Now, let's convert Option $2$ to exponential form. $\newline$ $(\sqrt[3]{y})^8 = (y^{1/3})^8$ $\newline$ Using the power to power rule of exponents, we multiply the exponents. $\newline$ $(y^{1/3})^8 = y^{1/3 \times 8} = y^{8/3}$ $\newline$ This is not equivalent to $y^{3/8}$ either.
Identify Mistake: Since neither of the given options is equivalent to $y^{\frac{3}{8}}$ , there seems to be a mistake in the options provided. We need to find a radical expression that correctly represents $y^{\frac{3}{8}}$ .
Express as Radical: To express $y^{\frac{3}{8}}$ as a single radical, we can write it as the $8$ th root of $y$ raised to the third power. $\newline$ $y^{\frac{3}{8}} = \sqrt[8]{y^3}$ $\newline$ This is the correct radical expression equivalent to $y^{\frac{3}{8}}$ .