Question

Emily claims that 3\frac{2}{3}\cdot\sqrt[6]{3^{3}}=3\frac{2}{3}\cdot3\frac{3}{6}=3\frac{6}{18}=\sqrt[3]{3}
Is Emily correct? Justify your answer

Answer 1

make it readable

Answer 2

To determine if Emily is correct, we will simplify both sides of the equation separately and compare the results.

Starting with the left side of the equation:
\begin{align*}
3\frac{2}{3}\cdot\sqrt[6]{3^{3}} &= \frac{11}{3} \cdot \sqrt[6]{3^3}
\end{align*}

We can simplify $\sqrt[6]{3^3}$ by raising $3$ to the power of $\frac{3}{6}$, which gives us $\sqrt[6]{3^3} = \sqrt[6]{3^2}$.

Now, let's simplify the expression further:
\begin{align*}
\frac{11}{3} \cdot \sqrt[6]{3^2} &= \frac{11}{3} \cdot \sqrt[6]{9} \\
&= \frac{11}{3} \cdot \sqrt[3]{\sqrt[6]{9}^2} \\
&= \frac{11}{3} \cdot \sqrt[3]{(\sqrt[6]{3^2})^2} \\
&= \frac{11}{3} \cdot \sqrt[3]{3^2} \\
&= \frac{11}{3} \cdot 3 \\
&= \frac{33}{3} \\
&= 11
\end{align*}

So, the left side simplifies to $11$.

Now, let's simplify the right side of the equation:
\begin{align*}
3\frac{2}{3}\cdot3\frac{3}{6} &= \frac{11}{3} \cdot \frac{11}{2} \\
&= \frac{11}{1} \cdot \frac{11}{1} \\
&= \frac{11 \cdot 11}{1 \cdot 1} \\
&= \frac{121}{1} \\
&= 121
\end{align*}

Therefore, the right side simplifies to $121$.

Since $11 \neq 121$, we can conclude that Emily's claim is incorrect. The two sides of the equation do not have equal values.

Answer 3

To determine if Emily is correct, we will simplify both sides of the equation separately and compare the results.

Starting with the left side of the equation:
\[3\frac{2}{3}\cdot\sqrt[6]{3^{3}} = \frac{11}{3} \cdot \sqrt[6]{3^3}\]

We can simplify $\sqrt[6]{3^3}$ as $\sqrt[6]{3^2}$.

Now, let's simplify the expression further:
\[\frac{11}{3} \cdot \sqrt[6]{3^2} = \frac{11}{3} \cdot \sqrt[3]{\sqrt[6]{9}^2} = \frac{11}{3} \cdot \sqrt[3]{(\sqrt[6]{3^2})^2} = \frac{11}{3} \cdot \sqrt[3]{3^2} = \frac{11}{3} \cdot 3 = \frac{33}{3} = 11\]

So, the left side simplifies to $11$.

Now, let's simplify the right side of the equation:
\[3\frac{2}{3}\cdot3\frac{3}{6} = \frac{11}{3} \cdot \frac{11}{2} = \frac{11}{1} \cdot \frac{11}{1} = \frac{11 \cdot 11}{1 \cdot 1} = \frac{121}{1} = 121\]

Therefore, the right side simplifies to $121$.

Since $11 \neq 121$, we can conclude that Emily's claim is incorrect. The two sides of the equation do not have equal values.