Certainly, let's break down the expression step-by-step:
Given expression:
[tex]\[ c \cdot \frac{\sqrt{9x}}{\sqrt[3]{3x}} \][/tex]
### Step 1: Simplify the square root and cube root individually
1. Square Root of [tex]\(9x\)[/tex]:
[tex]\[ \sqrt{9x} \][/tex]
Recall that the square root of a product can be rewritten as the product of the square roots:
[tex]\[ \sqrt{9x} = \sqrt{9} \cdot \sqrt{x} \][/tex]
Since the square root of 9 is 3:
[tex]\[ \sqrt{9x} = 3 \sqrt{x} \][/tex]
2. Cube Root of [tex]\(3x\)[/tex]:
[tex]\[ \sqrt[3]{3x} \][/tex]
Similarly, the cube root of a product can be rewritten as the product of the cube roots:
[tex]\[ \sqrt[3]{3x} = \sqrt[3]{3} \cdot \sqrt[3]{x} \][/tex]
### Step 2: Substitute back into the original expression
Substitute the simplified forms of [tex]\(\sqrt{9x}\)[/tex] and [tex]\(\sqrt[3]{3x}\)[/tex] into the given expression:
[tex]\[ c \cdot \frac{3 \sqrt{x}}{\sqrt[3]{3} \sqrt[3]{x}} \][/tex]
### Step 3: Simplify the division
The expression can be rewritten as:
[tex]\[ c \cdot \frac{3 \sqrt{x}}{\sqrt[3]{3} \cdot \sqrt[3]{x}} \][/tex]
Combine the radicals involving [tex]\(x\)[/tex]:
[tex]\[ c \cdot \frac{3 \cdot x^{1/2}}{3^{1/3} \cdot x^{1/3}} \][/tex]
Rewrite this as:
[tex]\[ c \cdot \frac{3 \cdot x^{1/2}}{3^{1/3} \cdot x^{1/3}} = c \cdot \frac{3}{3^{1/3}} \cdot \frac{x^{1/2}}{x^{1/3}} \][/tex]
Simplify the coefficients:
[tex]\[ \frac{3}{3^{1/3}} = 3^{1 - 1/3} = 3^{2/3} \][/tex]
Combine the exponents of [tex]\(x\)[/tex]:
[tex]\[ \frac{x^{1/2}}{x^{1/3}} = x^{1/2 - 1/3} = x^{3/6 - 2/6} = x^{1/6} \][/tex]
### Step 4: Combine all parts
Putting it all together, the expression simplifies to:
[tex]\[ c \cdot 3^{2/3} \cdot x^{1/6} \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ 3^{2/3} \cdot c \cdot x^{1/6} \][/tex]
