The value of `sqrt(a sqrt(b sqrt(c sqrt(d))))` is:

Options

A. `a^(1//2) b^(1//2) c^(1//2) d^(1//2)`
B. `a^(1//2) b^(1/2) c^(1//8) d^(1//16)`
C. `(abcd)^(1//12)`
D. `(abcd)^(1//8)`

Answer

To find the value of \( \sqrt{a \sqrt{b \sqrt{c \sqrt{d}}}} \), we can simplify the expression step by step.

### Step 1: Rewrite the square roots in exponential form
We know that \( \sqrt{x} = x^{1/2} \). Therefore, we can rewrite the expression as:
\[ \sqrt{a \sqrt{b \sqrt{c \sqrt{d}}}} = (a \sqrt{b \sqrt{c \sqrt{d}}})^{1/2} \]

### Step 2: Simplify the inner square root
Next, we simplify the inner square root \( \sqrt{b \sqrt{c \sqrt{d}}} \):
\[ \sqrt{b \sqrt{c \sqrt{d}}} = (b \sqrt{c \sqrt{d}})^{1/2} \]
Now, we can further simplify \( \sqrt{c \sqrt{d}} \):
\[ \sqrt{c \sqrt{d}} = (c \sqrt{d})^{1/2} \]
Continuing, we simplify \( \sqrt{d} \):
\[ \sqrt{d} = d^{1/2} \]
Thus, we have:
\[ \sqrt{c \sqrt{d}} = (c \cdot d^{1/2})^{1/2} = c^{1/2} \cdot d^{1/4} \]
Putting this back, we have:
\[ \sqrt{b \sqrt{c \sqrt{d}}} = (b \cdot c^{1/2} \cdot d^{1/4})^{1/2} = b^{1/2} \cdot c^{1/4} \cdot d^{1/8} \]

### Step 3: Substitute back into the original expression
Now substituting this back into our expression gives:
\[ \sqrt{a \sqrt{b \sqrt{c \sqrt{d}}}} = (a \cdot b^{1/2} \cdot c^{1/4} \cdot d^{1/8})^{1/2} \]

### Step 4: Apply the exponent
Using the property of exponents, we can distribute the \( 1/2 \):
\[ = a^{1/2} \cdot b^{1/4} \cdot c^{1/8} \cdot d^{1/16} \]

### Final Result
Thus, the value of \( \sqrt{a \sqrt{b \sqrt{c \sqrt{d}}}} \) is:
\[ a^{1/2} \cdot b^{1/4} \cdot c^{1/8} \cdot d^{1/16} \]