Simplify : `(25xx 216xx 729)/(6^(6)xx 5^(5)xx 9^(4))`

Options

A. `(1)/((30)^(7)(9)^(8))`
B. `(1)/((5)^(6)(6)^(9)(9)^(7))`
C. `(1)/((5)^(2)(6)^(3)(9)^(3))`
D. `(1)/((30)^(3) xx 9)`

Answer

To simplify the expression \((25 \times 216 \times 729)/(6^{6} \times 5^{5} \times 9^{4})\), we can follow these steps:

### Step 1: Factor the numbers in the numerator
- \(25 = 5^2\)
- \(216 = 6^3\)
- \(729 = 9^3\)

So, we can rewrite the numerator as:
\[ 25 \times 216 \times 729 = 5^2 \times 6^3 \times 9^3 \]

### Step 2: Rewrite \(9\) in terms of its prime factors
Since \(9 = 3^2\), we can express \(9^3\) as:
\[ 9^3 = (3^2)^3 = 3^6 \]

Thus, the numerator becomes:
\[ 5^2 \times 6^3 \times 3^6 \]

### Step 3: Rewrite \(6\) in terms of its prime factors
Since \(6 = 2 \times 3\), we can express \(6^3\) as:
\[ 6^3 = (2 \times 3)^3 = 2^3 \times 3^3 \]

Now substituting this back into the numerator gives:
\[ 5^2 \times (2^3 \times 3^3) \times 3^6 = 5^2 \times 2^3 \times 3^{3+6} = 5^2 \times 2^3 \times 3^9 \]

### Step 4: Write the denominator
The denominator is given as:
\[ 6^{6} \times 5^{5} \times 9^{4} \]

Now, we can rewrite \(6^6\) and \(9^4\):
- \(6^6 = (2 \times 3)^6 = 2^6 \times 3^6\)
- \(9^4 = (3^2)^4 = 3^8\)

Thus, the denominator becomes:
\[ (2^6 \times 3^6) \times 5^5 \times 3^8 = 2^6 \times 5^5 \times 3^{6+8} = 2^6 \times 5^5 \times 3^{14} \]

### Step 5: Combine the numerator and denominator
Now we can write the entire expression as:
\[ \frac{5^2 \times 2^3 \times 3^9}{2^6 \times 5^5 \times 3^{14}} \]

### Step 6: Simplify the expression
Now we can simplify by subtracting the exponents of the same bases:
- For \(2\): \(2^3 / 2^6 = 2^{3-6} = 2^{-3}\)
- For \(5\): \(5^2 / 5^5 = 5^{2-5} = 5^{-3}\)
- For \(3\): \(3^9 / 3^{14} = 3^{9-14} = 3^{-5}\)

So, the expression simplifies to:
\[ \frac{1}{2^3 \times 5^3 \times 3^5} \]

### Final Answer:
\[ \frac{1}{2^3 \times 5^3 \times 3^5} \]

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