`root3((-a^6 xx b^3 xx c^(21))/(c^9 xx a^(12)))=………`

Options

A. `(-bc^3)/(a^2)`
B. `(bc^4)/(a^2)`
C. `(-ab^4)/c^2`
D. `(-bc^4)/(a^2)`

Answer

To solve the expression \( \sqrt[3]{\frac{-a^6 \cdot b^3 \cdot c^{21}}{c^9 \cdot a^{12}}} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt[3]{\frac{-a^6 \cdot b^3 \cdot c^{21}}{c^9 \cdot a^{12}}} \]

### Step 2: Simplify the fraction inside the cube root
We can separate the numerator and the denominator:
\[ = \sqrt[3]{-a^6 \cdot b^3 \cdot c^{21}} \div \sqrt[3]{c^9 \cdot a^{12}} \]

### Step 3: Simplify the terms
Now we can simplify the terms:
\[ = \frac{\sqrt[3]{-a^6} \cdot \sqrt[3]{b^3} \cdot \sqrt[3]{c^{21}}}{\sqrt[3]{c^9} \cdot \sqrt[3]{a^{12}}} \]

### Step 4: Apply the cube root to each term
Using the property \( \sqrt[3]{x^n} = x^{n/3} \):
\[ = \frac{-a^{6/3} \cdot b^{3/3} \cdot c^{21/3}}{c^{9/3} \cdot a^{12/3}} \]

### Step 5: Simplify the powers
This simplifies to:
\[ = \frac{-a^2 \cdot b^1 \cdot c^7}{c^3 \cdot a^4} \]

### Step 6: Combine like terms
Now we can simplify the expression further:
\[ = \frac{-b \cdot c^7}{c^3 \cdot a^4} \cdot \frac{1}{a^2} \]
This gives us:
\[ = -b \cdot c^{7-3} \cdot a^{-4-2} \]
\[ = -b \cdot c^4 \cdot a^{-6} \]

### Step 7: Rewrite the expression
Finally, we can rewrite it as:
\[ = -\frac{b \cdot c^4}{a^6} \]

Thus, the final answer is:
\[ \sqrt[3]{\frac{-a^6 \cdot b^3 \cdot c^{21}}{c^9 \cdot a^{12}}} = -\frac{b \cdot c^4}{a^6} \]