We need to simplify the expression:
[tex]-2\sqrt[]{20}-2\sqrt[]{24}-2\sqrt[]{24}[/tex]We can start by grouping the second and third terms since they have the same factor:
[tex]-2\sqrt[]{24}-2\sqrt[]{24}=(-2-2)\sqrt[]{24}=-4\sqrt[]{24}[/tex]Then, we need to simplify:
[tex]-2\sqrt[]{20}-4\sqrt[]{24}[/tex]Now, we can factor the number inside each square root:
[tex]\begin{gathered} 20=2\cdot2\cdot5=2^2\cdot5 \\ \\ 24=2\cdot2\cdot2\cdot3=2^2\cdot6 \end{gathered}[/tex]And we can use the following properties:
[tex]\begin{gathered} \sqrt[]{a.b}=\sqrt[]{a}\cdot\sqrt[]{b} \\ \\ \sqrt[]{n^{2}}=n,\text{ for }n\ge0 \end{gathered}[/tex]Then, we obtain:
[tex]\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}[/tex]Using the above results in the expression, we find:
[tex]-2\sqrt[]{20}-4\sqrt[]{24}=-2\cdot2\sqrt[]{5}-4\cdot2\sqrt[]{6}=-4\sqrt[]{5}-4\cdot2\sqrt[]{6}[/tex]Since both terms have the factor -4, we can group it to obtain:
[tex]-4(\sqrt[]{5}+2\sqrt[]{6})[/tex]Therefore, a way to simplify the given expression is by writing it as:
[tex]-4(\sqrt[]{5}+2\sqrt[]{6})[/tex]