If`root3((a^(6) xx b^(3) xx c^(21))/(c^(9) xx a^(12)))=(bc^(k))/a^(k//2')`then k=______ .

Answer

To solve the equation

\[ \sqrt[3]{\frac{a^6 \cdot b^3 \cdot c^{21}}{c^9 \cdot a^{12}}} = \frac{b \cdot c^k}{a^{k/2}}, \]

we will follow these steps:

### Step 1: Simplify the left-hand side
We start by simplifying the expression inside the cube root:

\[ \frac{a^6 \cdot b^3 \cdot c^{21}}{c^9 \cdot a^{12}} = \frac{a^6}{a^{12}} \cdot \frac{b^3}{1} \cdot \frac{c^{21}}{c^9}. \]

Using the property of exponents \(\frac{m}{n} = m - n\), we can rewrite it as:

\[ = a^{6 - 12} \cdot b^3 \cdot c^{21 - 9} = a^{-6} \cdot b^3 \cdot c^{12}. \]

### Step 2: Apply the cube root
Now we take the cube root of the simplified expression:

\[ \sqrt[3]{a^{-6} \cdot b^3 \cdot c^{12}} = \sqrt[3]{a^{-6}} \cdot \sqrt[3]{b^3} \cdot \sqrt[3]{c^{12}}. \]

Calculating each term gives us:

\[ = a^{-6/3} \cdot b^{3/3} \cdot c^{12/3} = a^{-2} \cdot b^1 \cdot c^4 = \frac{b \cdot c^4}{a^2}. \]

### Step 3: Set the expressions equal
Now we set the left-hand side equal to the right-hand side:

\[ \frac{b \cdot c^4}{a^2} = \frac{b \cdot c^k}{a^{k/2}}. \]

### Step 4: Compare coefficients
Since the bases \(b\) are the same on both sides, we can cancel them out:

\[ \frac{c^4}{a^2} = \frac{c^k}{a^{k/2}}. \]

Now we can compare the coefficients of \(c\) and \(a\):

1. For \(c\): \(4 = k\)
2. For \(a\): \(2 = \frac{k}{2}\)

### Step 5: Solve for \(k\)
From the first equation, we have:

\[ k = 4. \]

From the second equation, multiplying both sides by 2 gives:

\[ k = 4. \]

### Conclusion
Thus, the value of \(k\) is

\[ \boxed{4}. \]