What is `5^(1/2) . 5^(1/4) .5^(1/8)…..` equal to

Options

A. 6
B. 1
C. 0
D. 5

Answer

To solve the expression \( 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8} \cdots \), we can follow these steps:

### Step 1: Identify the Pattern
The expression consists of a series of terms where the base is the same (5) and the exponents are fractions that follow a pattern. The exponents are \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \).

### Step 2: Use the Property of Exponents
When multiplying powers with the same base, we can add the exponents:
\[ 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8} = 5^{(1/2 + 1/4 + 1/8 + \ldots)} \]

### Step 3: Sum the Exponents
Now we need to find the sum of the series \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \). This is a geometric series where:
- The first term \( a = \frac{1}{2} \)
- The common ratio \( r = \frac{1/4}{1/2} = \frac{1}{2} \)

### Step 4: Use the Formula for the Sum of an Infinite Geometric Series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values we have:
\[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \]

### Step 5: Substitute Back into the Exponent
Now that we have the sum of the exponents, we substitute it back into the expression:
\[ 5^{(1/2 + 1/4 + 1/8 + \ldots)} = 5^1 \]

### Step 6: Final Result
Thus, the value of the original expression is:
\[ 5^1 = 5 \]

### Summary
The expression \( 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8} \cdots \) simplifies to \( 5 \).