Find the value of `(9^(3//2)-3xx5^(0)-[(1)/(81)]^(-1//2))/(((64)/(125))^(-2//3)+(1)/(((256)/(625))^(1//4))+((sqrt(25))/(root(3)(64))))` .

Options

A. `15//13`
B. `0`
C. `16//5`
D. `48//13`

Answer

To solve the expression

\[ \frac{9^{(3/2)} - 3 \cdot 5^{0} - \left(\frac{1}{81}\right)^{(-1/2)}}{\left(\frac{64}{125}\right)^{-2/3} + \frac{1}{\left(\frac{256}{625}\right)^{(1/4)}} + \left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)} \]

we will break it down step by step.

### Step 1: Simplify the Numerator

1. **Calculate \(9^{(3/2)}\)**:
\[ 9^{(3/2)} = (3^2)^{(3/2)} = 3^{(2 \cdot 3/2)} = 3^3 = 27 \]

2. **Calculate \(3 \cdot 5^{0}\)**:
\[ 5^{0} = 1 \quad \text{(any number to the power of 0 is 1)} \]
\[ 3 \cdot 5^{0} = 3 \cdot 1 = 3 \]

3. **Calculate \(\left(\frac{1}{81}\right)^{(-1/2)}\)**:
\[ 81 = 9^2 = (3^2)^2 = 3^4 \Rightarrow \frac{1}{81} = 3^{-4} \]
\[ \left(\frac{1}{81}\right)^{(-1/2)} = (3^{-4})^{(-1/2)} = 3^{(4/2)} = 3^2 = 9 \]

4. **Combine the results**:
\[ \text{Numerator} = 27 - 3 - 9 = 15 \]

### Step 2: Simplify the Denominator

1. **Calculate \(\left(\frac{64}{125}\right)^{-2/3}\)**:
\[ \frac{64}{125} = \left(\frac{4^3}{5^3}\right) \Rightarrow \left(\frac{64}{125}\right)^{-2/3} = \left(\frac{4}{5}\right)^{-2} = \left(\frac{5}{4}\right)^{2} = \frac{25}{16} \]

2. **Calculate \(\frac{1}{\left(\frac{256}{625}\right)^{(1/4)}}\)**:
\[ \frac{256}{625} = \left(\frac{16^2}{25^2}\right) \Rightarrow \left(\frac{256}{625}\right)^{(1/4)} = \frac{16^{1/2}}{25^{1/2}} = \frac{4}{5} \]
\[ \frac{1}{\left(\frac{256}{625}\right)^{(1/4)}} = \frac{5}{4} \]

3. **Calculate \(\left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)\)**:
\[ \sqrt{25} = 5 \quad \text{and} \quad \sqrt[3]{64} = 4 \]
\[ \frac{\sqrt{25}}{\sqrt[3]{64}} = \frac{5}{4} \]

4. **Combine the results**:
\[ \text{Denominator} = \frac{25}{16} + \frac{5}{4} + \frac{5}{4} \]
Convert \(\frac{5}{4}\) to have a common denominator of 16:
\[ \frac{5}{4} = \frac{20}{16} \]
So,
\[ \text{Denominator} = \frac{25}{16} + \frac{20}{16} + \frac{20}{16} = \frac{25 + 20 + 20}{16} = \frac{65}{16} \]

### Step 3: Combine Numerator and Denominator

Now we can write the overall expression:
\[ \frac{15}{\frac{65}{16}} = 15 \cdot \frac{16}{65} = \frac{240}{65} = \frac{48}{13} \]

### Final Answer
The value of the expression is:
\[ \frac{48}{13} \]