$$ \frac{2x^{\frac{1}{2}}.x^{\frac{2}{3}}.6x^{\frac{-7}{3}}}{9x^{\frac{-5}{3}}.x^{\frac{3}{2}}} $$

Question

UpStudy AI · Accepted Answer

Let's simplify the expression step by step:

$$
\frac{2x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} \cdot 6x^{\frac{-7}{3}}}{9x^{\frac{-5}{3}} \cdot x^{\frac{3}{2}}}
$$

### Step 1: Combine the Constants
Multiply the constants in the numerator and denominator:

- **Numerator constants:** $$2 \times 6 = 12$$
- **Denominator constants:** $$9$$

So the constants part becomes:

$$
\frac{12}{9} = \frac{4}{3}
$$

### Step 2: Combine the Exponents of $$x$$
Use the property of exponents $$x^a \cdot x^b = x^{a+b}$$ and $$\frac{x^a}{x^b} = x^{a-b}$$:

**Numerator exponents:**

$$
x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} \cdot x^{\frac{-7}{3}} = x^{\frac{1}{2} + \frac{2}{3} + \frac{-7}{3}} = x^{\frac{1}{2} + \frac{2 - 7}{3}} = x^{\frac{1}{2} - \frac{5}{3}} = x^{\frac{-7}{6}}
$$

**Denominator exponents:**

$$
x^{\frac{-5}{3}} \cdot x^{\frac{3}{2}} = x^{\frac{-5}{3} + \frac{3}{2}} = x^{\frac{-10 + 9}{6}} = x^{\frac{-1}{6}}
$$

**Combined exponents:**

$$
\frac{x^{\frac{-7}{6}}}{x^{\frac{-1}{6}}} = x^{\frac{-7}{6} - \left(\frac{-1}{6}\right)} = x^{\frac{-7 + 1}{6}} = x^{\frac{-6}{6}} = x^{-1}
$$

### Step 3: Combine Constants and Exponents
Now, combine the simplified constants and the exponent:

$$
\frac{4}{3} \cdot x^{-1} = \frac{4}{3x}
$$

### Final Simplified Expression

$$
\frac{4}{3x}
$$

**Answer:** 
$$
\frac{4}{3x}
$$
$$ \frac{4}{3x} $$

Upstudy AI Solution

Answer

Solution

Step 1: Combine the Constants

Step 2: Combine the Exponents of

Step 3: Combine Constants and Exponents

Final Simplified Expression

Beyond the Answer

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