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
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.
make it readable
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.