Question

Operations with Radical ExpressionsThis is a new topic that I’ve never done before so I’m not sure where to start

Answer 1

We need to simplify the expression:

`-2\sqrt[]{20}-2\sqrt[]{24}-2\sqrt[]{24}`

We can start by grouping the second and third terms since they have the same factor:

`-2\sqrt[]{24}-2\sqrt[]{24}=(-2-2)\sqrt[]{24}=-4\sqrt[]{24}`

Then, we need to simplify:

`-2\sqrt[]{20}-4\sqrt[]{24}`

Now, we can factor the number inside each square root:

`\begin{gathered} 20=2\cdot2\cdot5=2^2\cdot5 \\ \\ 24=2\cdot2\cdot2\cdot3=2^2\cdot6 \end{gathered}`

And we can use the following properties:

`\begin{gathered} \sqrt[]{a.b}=\sqrt[]{a}\cdot\sqrt[]{b} \\ \\ \sqrt[]{n^(2)}=n,\text{ for }n\ge0 \end{gathered}`

Then, we obtain:

`\begin{gathered} \sqrt[]{20}=\sqrt[]{2^2\cdot5}=\sqrt[]{2^(2)}\cdot\sqrt[]{5}=2\sqrt[]{5} \\ \\ \sqrt[]{24}=\sqrt[]{2^2\cdot6}=\sqrt[]{2^(2)}\cdot\sqrt[]{6}=2\sqrt[]{6} \end{gathered}`

Using the above results in the expression, we find:

`-2\sqrt[]{20}-4\sqrt[]{24}=-2\cdot2\sqrt[]{5}-4\cdot2\sqrt[]{6}=-4\sqrt[]{5}-4\cdot2\sqrt[]{6}`

Since both terms have the factor -4, we can group it to obtain:

`-4(\sqrt[]{5}+2\sqrt[]{6})`

Therefore, a way to simplify the given expression is by writing it as:

`-4(\sqrt[]{5}+2\sqrt[]{6})`