Combine these radicals. -12√12-2√3
The Correct Answer and Explanation is :
To combine the radicals in the expression (-12\sqrt{12} – 2\sqrt{3}), we first need to simplify (-12\sqrt{12}).
Step 1: Simplify (\sqrt{12})
The square root of 12 can be simplified as follows:
[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
]
Step 2: Substitute (\sqrt{12}) in the Expression
Now we can substitute this back into the expression:
[
-12\sqrt{12} = -12 \cdot 2\sqrt{3} = -24\sqrt{3}
]
Step 3: Rewrite the Expression
Now we can rewrite the original expression:
[
-12\sqrt{12} – 2\sqrt{3} = -24\sqrt{3} – 2\sqrt{3}
]
Step 4: Combine Like Terms
Since both terms now contain the same radical (\sqrt{3}), we can combine them:
[
-24\sqrt{3} – 2\sqrt{3} = (-24 – 2)\sqrt{3} = -26\sqrt{3}
]
Final Answer
Thus, the simplified expression is:
[
\boxed{-26\sqrt{3}}
]
Explanation
When combining radicals, it’s essential to first simplify any radicals that can be expressed in terms of simpler radicals. The square root of a product, like (\sqrt{12}), can be broken down using the property that (\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}). In this case, we used this property to simplify (\sqrt{12}) into (2\sqrt{3}).
Once we had (-12\sqrt{12}) simplified to (-24\sqrt{3}), we were able to see that both terms in the expression shared the radical (\sqrt{3}). This allowed us to combine them into a single term, which is a crucial technique when working with expressions involving radicals. By following these steps, we ensure the expression is simplified correctly and efficiently.