`(0.000729)^(-3//4)xx(0.09)^(-3//4)=`

Options

A. `10^(3)/3^(3)`
B. `10^(5)/3^(5)`
C. `10^(2)/3^(2)`
D. `10^(6)/3^(6)`

Answer

To solve the expression \((0.000729)^{-3/4} \times (0.09)^{-3/4}\), we can follow these steps:

### Step 1: Rewrite the numbers in fraction form
We can express \(0.000729\) and \(0.09\) as fractions:
\[ 0.000729 = \frac{729}{1000000} \quad \text{and} \quad 0.09 = \frac{9}{100} \]

### Step 2: Rewrite the expression
Now we can rewrite the expression:
\[ (0.000729)^{-3/4} \times (0.09)^{-3/4} = \left(\frac{729}{1000000}\right)^{-3/4} \times \left(\frac{9}{100}\right)^{-3/4} \]

### Step 3: Apply the power property
Using the property of exponents \((a/b)^n = a^n / b^n\), we can separate the fractions:
\[ = \frac{729^{-3/4}}{(1000000)^{-3/4}} \times \frac{9^{-3/4}}{(100)^{-3/4}} \]

### Step 4: Simplify the bases
Now let's simplify \(1000000\) and \(100\):
\[ 1000000 = 10^6 \quad \text{and} \quad 100 = 10^2 \]
So we can rewrite the expression as:
\[ = \frac{729^{-3/4} \times 9^{-3/4}}{(10^6)^{-3/4} \times (10^2)^{-3/4}} \]

### Step 5: Combine the powers
Now we can combine the powers:
\[ = \frac{(729 \times 9)^{-3/4}}{(10^6 \times 10^2)^{-3/4}} = \frac{(729 \times 9)^{-3/4}}{10^{(6+2)(-3/4)}} \]
\[ = \frac{(729 \times 9)^{-3/4}}{10^{-6}} \]

### Step 6: Calculate \(729 \times 9\)
Next, we calculate \(729 \times 9\):
\[ 729 = 27^2 = (3^3)^2 = 3^6 \quad \text{and} \quad 9 = 3^2 \]
Thus,
\[ 729 \times 9 = 3^6 \times 3^2 = 3^{6+2} = 3^8 \]

### Step 7: Substitute back into the expression
Now substituting back, we have:
\[ = \frac{(3^8)^{-3/4}}{10^{-6}} = \frac{3^{-6}}{10^{-6}} \]

### Step 8: Simplify the expression
This simplifies to:
\[ = 10^6 \times 3^{-6} = \frac{10^6}{3^6} \]

### Final Answer
Thus, the final answer is:
\[ \frac{10^6}{3^6} \]

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