Simplify: `([(8^(-3//4))^(5//2)]^(8//15) xx 16^(3//4))/(64^(-1//3))`

Options

A. `3/2`
B. `9/2`
C. 4
D. 1

Answer

To simplify the expression \(\frac{([(8^{-3/4})^{(5/2)}]^{(8/15)} \times 16^{(3/4)})}{64^{-1/3}}\), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
- We know that:
- \(8 = 2^3\)
- \(16 = 2^4\)
- \(64 = 2^6\)

So we can rewrite the expression as:
\[ \frac{([(2^3)^{-3/4}]^{(5/2)}]^{(8/15)} \times (2^4)^{(3/4)})}{(2^6)^{-1/3}} \]

### Step 2: Simplify the powers
Using the power of a power property \((a^m)^n = a^{m \cdot n}\):
- For \(8^{-3/4}\):
\[ (2^3)^{-3/4} = 2^{(-3/4) \cdot 3} = 2^{-9/4} \]
- Now raise this to the power of \(5/2\):
\[ (2^{-9/4})^{(5/2)} = 2^{(-9/4) \cdot (5/2)} = 2^{-45/8} \]
- Now raise this to the power of \(8/15\):
\[ (2^{-45/8})^{(8/15)} = 2^{(-45/8) \cdot (8/15)} = 2^{-3/5} \]

### Step 3: Simplify \(16^{(3/4)}\)
Using the same power of a power property:
\[ (2^4)^{(3/4)} = 2^{(4 \cdot 3/4)} = 2^3 \]

### Step 4: Combine the results
Now we can combine the results:
\[ 2^{-3/5} \times 2^3 = 2^{-3/5 + 3} = 2^{-3/5 + 15/5} = 2^{12/5} \]

### Step 5: Simplify the denominator
Now simplify the denominator:
\[ (2^6)^{-1/3} = 2^{-6 \cdot (1/3)} = 2^{-2} \]

### Step 6: Combine the numerator and denominator
Now we have:
\[ \frac{2^{12/5}}{2^{-2}} = 2^{12/5 - (-2)} = 2^{12/5 + 2} = 2^{12/5 + 10/5} = 2^{22/5} \]

### Step 7: Final answer
The simplified form of the expression is:
\[ 2^{22/5} \]

### Step 8: Convert to a numerical value (optional)
If needed, we can express \(2^{22/5}\) as:
\[ 2^{22/5} = \sqrt[5]{2^{22}} = \sqrt[5]{4^5 \cdot 2^2} = 4 \cdot \sqrt[5]{4} \]