A) Describe the two errors in the students work simplifying the radical expression
The first thing we have to do is take the factors of the number
![\begin{gathered} \sqrt[3]{108\cdot a^(11)\cdot b^(14)\cdot c^5} \\ 108=2^2\cdot3^3 \\ \sqrt[3]{2^2\cdot3^3\cdot a^(11)\cdot b^(14)\cdot c^5} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/y6edn1f1y0f7tk876fjnis69tg9s3m9ajm.png)
• The first error is the factors of 108 missing to square the 2
Now we decompose each of the variables into factors of 3:

We add the factors within the cubic root
![\sqrt[3]{2^2\cdot3^3\cdot a^9\cdot a^2\cdot b^(12)\cdot b^2\cdot c^3\cdot c^2^{}}](https://img.qammunity.org/2023/formulas/mathematics/college/2rsty4x5s2z1cxntxbrw9161cm0jme16rh.png)
• The second error is the factors of the variable b because raised to 14 is b^12+ b^2 squared not b^9 +b^2
To finish we solve by taking all the factors cubed:
![\begin{gathered} \sqrt[3]{2^2\cdot3^3\cdot a^9\cdot a^2\cdot b^(12)\cdot b^2\cdot c^3\cdot c^2} \\ 3\cdot a^3\cdot b^4\cdot c\cdot\sqrt[3]{2^2\cdot a^2\cdot b^2\cdot c^2} \\ 3\cdot a^3\cdot b^4\cdot c\cdot\sqrt[3]{4^{}\cdot a^2\cdot b^2\cdot c^2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/cetl4ot4etmxbu8nhkun2hpce6oyt9k8xx.png)
So the answer is
![3\cdot a^3\cdot b^4\cdot c\cdot\sqrt[3]{4^{}\cdot a^2\cdot b^2\cdot c^2}](https://img.qammunity.org/2023/formulas/mathematics/college/5kyyknxwsehpm6sk7vrpy9mfn6kxovwa12.png)