If `sqrt (7 sqrt (7 sqrt7 sqrt7 sqrt7)) = 7 ^(x) ` then find the value of x

Options

A. `1/5`
B. `(32)/(31)`
C. `(9)/(8)`
D. `(1)/(32)`

Answer

To solve the equation \( \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7}}}}} = 7^x \), we will simplify the left-hand side step by step.

### Step 1: Rewrite the expression
We can start by rewriting the nested square roots. The expression can be simplified as follows:
\[ \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7}}}}} = \sqrt{7 \cdot \sqrt{7 \cdot \sqrt{7 \cdot \sqrt{7 \cdot \sqrt{7}}}}} \]

### Step 2: Simplify the innermost square root
Let’s denote the innermost square root:
\[ y = \sqrt{7} \]
Then,
\[ \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7}}}} = \sqrt{7 \cdot \sqrt{7 \cdot \sqrt{7 \cdot y}}} \]

### Step 3: Continue simplifying
Continuing this process, we can express the square roots in terms of powers of 7:
\[ \sqrt{7} = 7^{1/2} \]
Thus, we can express the entire nested square root as:
\[ \sqrt{7 \cdot 7^{1/2} \cdot 7^{1/4} \cdot 7^{1/8} \cdots} \]

### Step 4: Convert all square roots to exponents
Each square root can be expressed as:
\[ \sqrt{7} = 7^{1/2}, \quad \sqrt{7 \sqrt{7}} = 7^{1/2 + 1/4}, \quad \sqrt{7 \sqrt{7 \sqrt{7}}} = 7^{1/2 + 1/4 + 1/8}, \ldots \]

### Step 5: Sum the exponents
The exponents form a geometric series:
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \]
This series can be summed up using the formula for the sum of an infinite geometric series:
\[ S = \frac{a}{1 - r} \]
where \( a = \frac{1}{2} \) and \( r = \frac{1}{2} \):
\[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \]

### Step 6: Combine the results
Thus, the total exponent is:
\[ \sqrt{7^{1 + 1}} = \sqrt{7^1} = 7^{1/2} \]

### Step 7: Final simplification
Now, we can express the entire left-hand side as:
\[ \sqrt{7^{\frac{9}{4}}} = 7^{\frac{9}{8}} \]

### Step 8: Equate the powers
Now we have:
\[ 7^{\frac{9}{8}} = 7^x \]
Since the bases are the same, we can equate the exponents:
\[ x = \frac{9}{8} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{9}{8}} \]