Question

A) Describe the two errors in the students work simplifying the radical expression B) correctly simplify the radical expression

Answer 1

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}`

• The first error is the factors of 108 missing to square the 2

Now we decompose each of the variables into factors of 3:

`\begin{gathered} a^(11)=a^3\cdot a^3\cdot a^3\cdot a^2=(a^3)^3\cdot a^2=a^9\cdot a^2 \\ b^(14)=b^3\cdot b^3\cdot b^3\cdot b^3\cdot b^2=(b^3)^4\cdot b^2=b^(12)\cdot b^2 \\ c^5=c^3\cdot c^2 \end{gathered}`

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^{}}`

• 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}`

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}`