Sure, let's simplify the expression [tex]\( 4x\sqrt{3x} - x\sqrt{3x} - 2x\sqrt{3x} \)[/tex] step-by-step.
1. Identify the terms: The expression consists of three terms that all share a common factor. Let's list them individually:
[tex]\[
4x\sqrt{3x}, \quad -x\sqrt{3x}, \quad -2x\sqrt{3x}
\][/tex]
2. Combine like terms: All three terms have the common factor [tex]\( x\sqrt{3x} \)[/tex]. We can factor this out to combine them:
[tex]\[
(4x - x - 2x)\sqrt{3x}
\][/tex]
3. Simplify the coefficients: Add the coefficients of [tex]\( x\sqrt{3x} \)[/tex]:
[tex]\[
4 - 1 - 2
\][/tex]
Calculate the sum of these coefficients:
[tex]\[
4 - 1 - 2 = 1
\][/tex]
4. Substitute the simplified coefficient back in: Now that we have simplified the coefficients, substitute this back into the expression:
[tex]\[
1 \cdot x\sqrt{3x} = x\sqrt{3x}
\][/tex]
5. Express the result: The expression [tex]\( x\sqrt{3x} \)[/tex] can be further simplified using the property of exponents. Note that
[tex]\[
\sqrt{3x} = (3x)^{1/2}
\][/tex]
Therefore,
[tex]\[
x\sqrt{3x} = x \cdot (3x)^{1/2} = x \cdot 3^{1/2} \cdot x^{1/2} = \sqrt{3} \cdot x \cdot x^{1/2} = \sqrt{3} \cdot x^{1 + 1/2} = \sqrt{3} \cdot x^{3/2}
\][/tex]
So, the simplified form of the expression [tex]\( 4x\sqrt{3x} - x\sqrt{3x} - 2x\sqrt{3x} \)[/tex] is:
[tex]\[
\sqrt{3}x^{3/2}
\][/tex]
Thus, the final result is [tex]\(\sqrt{3}x^{3/2}\)[/tex].
