Simplify [tex]$4x \sqrt{3x} - x \sqrt{3x} - 2x \sqrt{3x}$[/tex].

A. [tex]x \sqrt{3x}[/tex]
B. [tex]x \sqrt{9x}[/tex]
C. [tex]2x \sqrt{6x}[/tex]
D. [tex]2x \sqrt{6x^3}[/tex]



Answer :

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].