If `k ne 0`, what is the value of `(9(2k)^(3))/((3k)^(3))`?

Answer

To solve the expression \((9(2k)^{3})/((3k)^{3})\), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \frac{9(2k)^{3}}{(3k)^{3}} \]

### Step 2: Expand the cubes
Using the property of exponents \((AB)^{n} = A^{n}B^{n}\), we can expand both the numerator and the denominator:
\[ (2k)^{3} = 2^{3}k^{3} = 8k^{3} \]
\[ (3k)^{3} = 3^{3}k^{3} = 27k^{3} \]
Now substituting these back into the expression, we have:
\[ \frac{9 \cdot 8k^{3}}{27k^{3}} \]

### Step 3: Simplify the expression
Now we can simplify the expression. The \(k^{3}\) in the numerator and denominator cancels out:
\[ \frac{9 \cdot 8}{27} \]

### Step 4: Calculate the numerical values
Now we calculate \(9 \cdot 8\) and then divide by \(27\):
\[ 9 \cdot 8 = 72 \]
So, we have:
\[ \frac{72}{27} \]

### Step 5: Simplify the fraction
We can simplify \(\frac{72}{27}\) by finding the greatest common divisor (GCD) of \(72\) and \(27\). The GCD is \(9\):
\[ \frac{72 \div 9}{27 \div 9} = \frac{8}{3} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{8}{3} \]

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