`([(625)^(1//2) xx (1024)^(1//5) ]^(1//2))/([(root(4)(256))^(1//2)] xx (125)^(1//3))` equals

Options

A. `1/5`
B. `1/(125)`
C. 1
D. `1/10`

Answer

To solve the expression \(\frac{([(625)^{\frac{1}{2}} \times (1024)^{\frac{1}{5}}]^{\frac{1}{2}})}{([( \sqrt[4]{256})^{\frac{1}{2}}] \times (125)^{\frac{1}{3}})}\), we will simplify it step by step.

### Step 1: Simplify \(625\) and \(1024\)
- \(625 = 25^2 = (5^2)^2 = 5^4\)
- \(1024 = 2^{10}\)

### Step 2: Rewrite the expression
Now, substituting these values into the expression, we have:
\[ \frac{([(5^4)^{\frac{1}{2}} \times (2^{10})^{\frac{1}{5}}]^{\frac{1}{2}})}{([( \sqrt[4]{256})^{\frac{1}{2}}] \times (125)^{\frac{1}{3}})} \]

### Step 3: Simplify the powers
- For \(625\):
\[ (5^4)^{\frac{1}{2}} = 5^{4 \times \frac{1}{2}} = 5^2 \]
- For \(1024\):
\[ (2^{10})^{\frac{1}{5}} = 2^{10 \times \frac{1}{5}} = 2^2 \]

### Step 4: Combine the results
Now we combine these results:
\[ 5^2 \times 2^2 = 25 \times 4 = 100 \]

### Step 5: Apply the outer power
Now we apply the outer power:
\[ (100)^{\frac{1}{2}} = \sqrt{100} = 10 \]

### Step 6: Simplify the denominator
Now let's simplify the denominator:
- \(256 = 2^8\), so:
\[ \sqrt[4]{256} = (2^8)^{\frac{1}{4}} = 2^{8 \times \frac{1}{4}} = 2^2 = 4 \]
- For \(125\):
\[ 125 = 5^3 \implies (125)^{\frac{1}{3}} = (5^3)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}} = 5^1 = 5 \]

### Step 7: Combine the denominator
Now we combine these results:
\[ 4^{\frac{1}{2}} \times 5 = 2 \times 5 = 10 \]

### Step 8: Final Calculation
Now we have the final expression:
\[ \frac{10}{10} = 1 \]

### Final Answer
Hence, the required answer is:
\[ \boxed{1} \]