Answer :
To combine the given radicals, we need to handle like terms and simplify the expressions where possible. Here's the step-by-step solution:
### 1. Identify and group like terms
First, observe that we have radicals involving [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{12}\)[/tex]. Notice that [tex]\(\sqrt{12}\)[/tex] can be rewritten as [tex]\(\sqrt{4 \cdot 3} = 2\sqrt{3}\)[/tex]. By doing this, we can transform all terms into expressions involving [tex]\(\sqrt{3}\)[/tex].
### 2. Convert [tex]\(\sqrt{12}\)[/tex] terms to [tex]\(\sqrt{3}\)[/tex]
- The term [tex]\( -12 \sqrt{12} \)[/tex] becomes:
[tex]\[ -12 \sqrt{12} = -12 \times 2 \sqrt{3} = -24 \sqrt{3} \][/tex]
- The term [tex]\( -10 \sqrt{12} \)[/tex] becomes:
[tex]\[ -10 \sqrt{12} = -10 \times 2 \sqrt{3} = -20 \sqrt{3} \][/tex]
### 3. Write all the terms in terms of [tex]\(\sqrt{3}\)[/tex]
Collectively, we have:
[tex]\[ -24 \sqrt{3}, \quad -2 \sqrt{3}, \quad -50 \sqrt{3}, \quad -22 \sqrt{3}, \quad -26 \sqrt{3}, \quad -20 \sqrt{3} \][/tex]
### 4. Combine like terms
Sum all the coefficients of [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ -24 - 2 - 50 - 22 - 26 - 20 = -144 \][/tex]
Therefore, the combined radicals are:
[tex]\[ -144 \sqrt{3} \][/tex]
### Conclusion
The combined radical expression is:
[tex]\[ -144 \sqrt{3} \][/tex]
This is the final simplified form of the given radicals when combined.
### 1. Identify and group like terms
First, observe that we have radicals involving [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{12}\)[/tex]. Notice that [tex]\(\sqrt{12}\)[/tex] can be rewritten as [tex]\(\sqrt{4 \cdot 3} = 2\sqrt{3}\)[/tex]. By doing this, we can transform all terms into expressions involving [tex]\(\sqrt{3}\)[/tex].
### 2. Convert [tex]\(\sqrt{12}\)[/tex] terms to [tex]\(\sqrt{3}\)[/tex]
- The term [tex]\( -12 \sqrt{12} \)[/tex] becomes:
[tex]\[ -12 \sqrt{12} = -12 \times 2 \sqrt{3} = -24 \sqrt{3} \][/tex]
- The term [tex]\( -10 \sqrt{12} \)[/tex] becomes:
[tex]\[ -10 \sqrt{12} = -10 \times 2 \sqrt{3} = -20 \sqrt{3} \][/tex]
### 3. Write all the terms in terms of [tex]\(\sqrt{3}\)[/tex]
Collectively, we have:
[tex]\[ -24 \sqrt{3}, \quad -2 \sqrt{3}, \quad -50 \sqrt{3}, \quad -22 \sqrt{3}, \quad -26 \sqrt{3}, \quad -20 \sqrt{3} \][/tex]
### 4. Combine like terms
Sum all the coefficients of [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ -24 - 2 - 50 - 22 - 26 - 20 = -144 \][/tex]
Therefore, the combined radicals are:
[tex]\[ -144 \sqrt{3} \][/tex]
### Conclusion
The combined radical expression is:
[tex]\[ -144 \sqrt{3} \][/tex]
This is the final simplified form of the given radicals when combined.
