To simplify the expression [tex]\( 4x\sqrt{3x} - x\sqrt{3x} - 2x\sqrt{3x} \)[/tex], we need to combine like terms. Here is the detailed, step-by-step solution:
1. Identify the terms:
- The first term is [tex]\( 4x\sqrt{3x} \)[/tex].
- The second term is [tex]\( -x\sqrt{3x} \)[/tex].
- The third term is [tex]\( -2x\sqrt{3x} \)[/tex].
2. Express each term in a more general form:
- Notice that all the terms contain the common factor [tex]\( x\sqrt{3x} \)[/tex], which we can express as [tex]\( \sqrt{3} \cdot x^{3/2} \)[/tex]. This simplifies our notation.
3. Rewrite each term with the common factor:
- The first term [tex]\( 4x\sqrt{3x} \)[/tex] can be written as [tex]\( 4 \cdot (\sqrt{3} \cdot x^{3/2}) \)[/tex], which simplifies to [tex]\( 4\sqrt{3} \cdot x^{3/2} \)[/tex].
- The second term [tex]\( -x\sqrt{3x} \)[/tex] can be written as [tex]\( -(\sqrt{3} \cdot x^{3/2}) \)[/tex], which simplifies to [tex]\( -\sqrt{3} \cdot x^{3/2} \)[/tex].
- The third term [tex]\( -2x\sqrt{3x} \)[/tex] can be written as [tex]\( -2 \cdot (\sqrt{3} \cdot x^{3/2}) \)[/tex], which simplifies to [tex]\( -2\sqrt{3} \cdot x^{3/2} \)[/tex].
4. Combine the terms:
- Now we have the expression [tex]\( 4\sqrt{3} \cdot x^{3/2} - \sqrt{3} \cdot x^{3/2} - 2\sqrt{3} \cdot x^{3/2} \)[/tex].
- We can factor out the common term [tex]\( \sqrt{3} \cdot x^{3/2} \)[/tex], giving us:
[tex]\[
(4 - 1 - 2) \cdot \sqrt{3} \cdot x^{3/2}
\][/tex]
5. Perform the arithmetic:
- Simplify the coefficients: [tex]\( 4 - 1 - 2 = 1 \)[/tex].
6. Combine the simplified coefficients with the common factor:
- We are left with:
[tex]\[
1 \cdot \sqrt{3} \cdot x^{3/2} = \sqrt{3} \cdot x^{3/2}
\][/tex]
7. Write the final simplified expression:
- Therefore, the simplified expression is:
[tex]\[
\sqrt{3} x^{3/2}
\][/tex]
In conclusion, the expression [tex]\( 4x\sqrt{3x} - x\sqrt{3x} - 2x\sqrt{3x} \)[/tex] simplifies to [tex]\( \sqrt{3} x^{3/2} \)[/tex].
