Certainly! Let's combine and simplify the given expressions step-by-step.
### Step 1: Identify Given Expressions
We have two expressions:
[tex]\[ 3 \sqrt{27 x^3} \][/tex]
[tex]\[ 2 \sqrt{12 x^3} \][/tex]
### Step 2: Simplify Each Expression Individually
#### For [tex]\( 3 \sqrt{27 x^3} \)[/tex]:
Rewrite the term under the square root:
[tex]\[ 3 \sqrt{27 x^3} = 3 \sqrt{3^3 \cdot 3 x^3} = 3 \sqrt{(3 \sqrt{x})^2 \cdot 3} = 3 \cdot 3 \sqrt{3} \sqrt{x^3} \][/tex]
Now simplify it:
[tex]\[ 3 \cdot 3 \sqrt{3} \cdot \sqrt{x^3} = 9 \sqrt{3} \cdot \sqrt{x^3} \][/tex]
#### For [tex]\( 2 \sqrt{12 x^3} \)[/tex]:
Rewrite the term under the square root:
[tex]\[ 2 \sqrt{12 x^3} = 2 \sqrt{4 \cdot 3 \cdot x^3} = 2 \sqrt{(2 \sqrt{x})^2 \cdot 3} = 2 \cdot 2 \sqrt{3} \cdot \sqrt{x^3} \][/tex]
Now simplify it:
[tex]\[ 2 \cdot 2 \sqrt{3} \cdot \sqrt{x^3} = 4 \sqrt{3} \cdot \sqrt{x^3} \][/tex]
### Step 3: Combine the Expressions
Now that we have simplified both terms, we can combine them:
[tex]\[ 3 \sqrt{27 x^3} - 2 \sqrt{12 x^3} = 9 \sqrt{3} \sqrt{x^3} - 4 \sqrt{3} \sqrt{x^3} \][/tex]
### Step 4: Simplify the Combined Expression
Both terms contain a common factor [tex]\( \sqrt{3} \sqrt{x^3} \)[/tex]:
[tex]\[ (9 \sqrt{3} \sqrt{x^3}) - (4 \sqrt{3} \sqrt{x^3}) = (9 - 4) \sqrt{3} \sqrt{x^3} \][/tex]
Combine the constants:
[tex]\[ (9 - 4) \sqrt{3} \sqrt{x^3} = 5 \sqrt{3} \sqrt{x^3} \][/tex]
### Conclusion
So, the simplified combination of the given expressions is:
[tex]\[ 3 \sqrt{27 x^3} - 2 \sqrt{12 x^3} = 5 \sqrt{3} \sqrt{x^3} \][/tex]
This is the final answer.
