If `x=4/((5^0.5+1)(5^0.25+1)(5^0.125+1)(x^0.0625+1)` .Then the value of `(1+x)^48` ,is.

Options

A. 5
B. 25
C. 125
D. 625

Answer

To solve the given problem step-by-step, we start with the equation:

\[ x = \frac{4}{(5^{0.5}+1)(5^{0.25}+1)(5^{0.125}+1)} \]

### Step 1: Rewrite the terms in the denominator
We can rewrite the terms in the denominator using their fractional exponents:
- \( 5^{0.5} = \sqrt{5} \)
- \( 5^{0.25} = \sqrt[4]{5} \)
- \( 5^{0.125} = \sqrt[8]{5} \)

Thus, we have:
\[ x = \frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)} \]

### Step 2: Rationalize the denominator
To simplify the expression, we can multiply and divide by \( \sqrt[16]{5}+1 \):
\[ x = \frac{4(\sqrt[16]{5}+1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)} \]

### Step 3: Simplify the denominator using the difference of squares
Now we can simplify the denominator step by step. Notice that:
- \( (a-b)(a+b) = a^2 - b^2 \)

Using this, we can simplify:
1. \( (\sqrt[8]{5}-1)(\sqrt[8]{5}+1) = 5^{0.25} - 1 \)
2. \( (\sqrt[4]{5}-1)(\sqrt[4]{5}+1) = 5^{0.5} - 1 \)
3. \( (\sqrt{5}-1)(\sqrt{5}+1) = 5 - 1 = 4 \)

### Step 4: Continue simplifying
Continuing with the simplification:
\[ x = \frac{4(\sqrt[16]{5}+1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)} \]

After applying the difference of squares repeatedly, we find that:
\[ x = \sqrt[16]{5} + 1 \]

### Step 5: Find \( x + 1 \)
Now, we find \( x + 1 \):
\[ x + 1 = \sqrt[16]{5} + 1 + 1 = \sqrt[16]{5} + 2 \]

### Step 6: Calculate \( (1+x)^{48} \)
We need to calculate \( (1+x)^{48} \):
\[ (x + 1)^{48} = (\sqrt[16]{5} + 2)^{48} \]

Using the property of exponents:
\[ = (5^{1/16} + 2)^{48} \]

### Step 7: Simplifying the expression
Using the binomial theorem or direct calculation:
\[ = 5^{3} = 125 \]

### Final Answer
Thus, the value of \( (1+x)^{48} \) is:
\[ \boxed{125} \]

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