The value of `((5)^(0.25)xx(125)^(0.25))/((256)^(0.10)xx(256)^(0.15))` is

Options

A. `(sqrt(5))/(2)`
B. `(5)/(4)`
C. `(25)/(2)`
D. `(25)/(16)`

Answer

To solve the expression \(\frac{(5^{0.25} \times 125^{0.25})}{(256^{0.10} \times 256^{0.15})}\), we can follow these steps:

### Step 1: Simplify the numerator
The numerator is \(5^{0.25} \times 125^{0.25}\). Since both terms have the same exponent, we can combine them:
\[ 5^{0.25} \times 125^{0.25} = (5 \times 125)^{0.25} \]
Now, we know that \(125 = 5^3\), so:
\[ 5 \times 125 = 5 \times 5^3 = 5^4 \]
Thus, we can rewrite the numerator as:
\[ (5^4)^{0.25} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \times n}\), we have:
\[ (5^4)^{0.25} = 5^{4 \times 0.25} = 5^{1} = 5 \]

### Step 3: Simplify the denominator
The denominator is \(256^{0.10} \times 256^{0.15}\). Again, since both terms have the same base, we can combine them:
\[ 256^{0.10} \times 256^{0.15} = 256^{0.10 + 0.15} = 256^{0.25} \]

### Step 4: Rewrite 256 in terms of powers of 4
We know that \(256 = 4^4\). Therefore:
\[ 256^{0.25} = (4^4)^{0.25} \]

### Step 5: Apply the power of a power rule again
Using the power of a power rule:
\[ (4^4)^{0.25} = 4^{4 \times 0.25} = 4^{1} = 4 \]

### Step 6: Combine the results
Now we can substitute back into our original expression:
\[ \frac{5}{4} \]

### Final Answer
Thus, the value of the expression \(\frac{(5^{0.25} \times 125^{0.25})}{(256^{0.10} \times 256^{0.15})}\) is:
\[ \frac{5}{4} \]