`root3((3^6 xx 4^3 xx 2^6)/(8^9 xx 2^3))=……….`

Options

A. `3/8`
B. `9/8`
C. `3/(64)`
D. `9/(64)`

Answer

To solve the expression \( \sqrt[3]{\frac{3^6 \times 4^3 \times 2^6}{8^9 \times 2^3}} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt[3]{\frac{3^6 \times 4^3 \times 2^6}{8^9 \times 2^3}} \]

### Step 2: Simplify the base of 8
We know that \( 8 = 2^3 \). Therefore, we can rewrite \( 8^9 \) as:
\[ 8^9 = (2^3)^9 = 2^{27} \]
Now, substituting this back into the expression, we have:
\[ \sqrt[3]{\frac{3^6 \times 4^3 \times 2^6}{2^{27} \times 2^3}} \]

### Step 3: Combine the powers of 2 in the denominator
The denominator can be simplified:
\[ 2^{27} \times 2^3 = 2^{27 + 3} = 2^{30} \]
So, the expression now looks like:
\[ \sqrt[3]{\frac{3^6 \times 4^3 \times 2^6}{2^{30}}} \]

### Step 4: Rewrite 4 in terms of 2
Since \( 4 = 2^2 \), we can rewrite \( 4^3 \) as:
\[ 4^3 = (2^2)^3 = 2^6 \]
Substituting this back into the expression gives us:
\[ \sqrt[3]{\frac{3^6 \times 2^6 \times 2^6}{2^{30}}} \]

### Step 5: Combine the powers of 2 in the numerator
Now, we can combine the powers of 2 in the numerator:
\[ 2^6 \times 2^6 = 2^{6 + 6} = 2^{12} \]
Thus, the expression simplifies to:
\[ \sqrt[3]{\frac{3^6 \times 2^{12}}{2^{30}}} \]

### Step 6: Simplify the fraction
Now, we can simplify the fraction:
\[ \frac{2^{12}}{2^{30}} = 2^{12 - 30} = 2^{-18} \]
So the expression now looks like:
\[ \sqrt[3]{3^6 \times 2^{-18}} \]

### Step 7: Apply the cube root
We can apply the cube root to both parts of the expression:
\[ \sqrt[3]{3^6} \times \sqrt[3]{2^{-18}} = 3^{6/3} \times 2^{-18/3} = 3^2 \times 2^{-6} \]

### Step 8: Final simplification
Now, we can express this as:
\[ 3^2 \times \frac{1}{2^6} = \frac{3^2}{2^6} = \frac{9}{64} \]

### Final Answer
Thus, the final answer is:
\[ \frac{9}{64} \]

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