The value of `(0.000729)^(-3//4)xx(0.09)^(-3//4)` is:

Options

A. `10000/81`
B. `1000000/729`
C. `729/1000000`
D. `81/100000`

Answer

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

### Step 1: Rewrite the numbers in terms of powers
First, we can express \(0.000729\) and \(0.09\) as powers of \(0.09\):

1. **Identify \(0.000729\)**:
\[ 0.000729 = 0.09^3 \]
This is because \(0.09 = \frac{9}{100}\) and when cubed, it gives \(0.000729\).

2. **Identify \(0.09\)**:
\[ 0.09 = 0.09^1 \]

### Step 2: Substitute the powers into the expression
Now we can substitute these values into the original expression:
\[ (0.000729)^{-\frac{3}{4}} = (0.09^3)^{-\frac{3}{4}} \]
Using the power of a power property, we multiply the exponents:
\[ (0.09^3)^{-\frac{3}{4}} = 0.09^{3 \times -\frac{3}{4}} = 0.09^{-\frac{9}{4}} \]

Now, substituting this back into the expression:
\[ (0.000729)^{-\frac{3}{4}} \times (0.09)^{-\frac{3}{4}} = 0.09^{-\frac{9}{4}} \times 0.09^{-\frac{3}{4}} \]

### Step 3: Combine the exponents
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\):
\[ 0.09^{-\frac{9}{4} - \frac{3}{4}} = 0.09^{-\frac{12}{4}} = 0.09^{-3} \]

### Step 4: Rewrite the negative exponent
A negative exponent indicates the reciprocal:
\[ 0.09^{-3} = \frac{1}{0.09^3} \]

### Step 5: Calculate \(0.09^3\)
Now we calculate \(0.09^3\):
\[ 0.09 = \frac{9}{100} \implies 0.09^3 = \left(\frac{9}{100}\right)^3 = \frac{9^3}{100^3} = \frac{729}{1000000} \]

### Step 6: Find the reciprocal
Now we can find the reciprocal:
\[ \frac{1}{0.09^3} = \frac{1}{\frac{729}{1000000}} = \frac{1000000}{729} \]

### Final Answer
Thus, the final value of \((0.000729)^{-\frac{3}{4}} \times (0.09)^{-\frac{3}{4}}\) is:
\[ \frac{1000000}{729} \]

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