Prove that
If ` (1)/(9!) +(1)/( 10!) =(x)/( 11!) ,` Find x.

Answer

To solve the equation \( \frac{1}{9!} + \frac{1}{10!} = \frac{x}{11!} \) and find the value of \( x \), we can follow these steps:

### Step 1: Write the equation
We start with the equation:
\[ \frac{1}{9!} + \frac{1}{10!} = \frac{x}{11!} \]

### Step 2: Find a common denominator
The common denominator for \( 9! \) and \( 10! \) is \( 10! \). We can rewrite \( \frac{1}{9!} \) as:
\[ \frac{1}{9!} = \frac{10}{10!} \]
Thus, we can rewrite the left-hand side:
\[ \frac{10}{10!} + \frac{1}{10!} = \frac{10 + 1}{10!} = \frac{11}{10!} \]

### Step 3: Substitute back into the equation
Now we substitute this back into the original equation:
\[ \frac{11}{10!} = \frac{x}{11!} \]

### Step 4: Express \( 11! \) in terms of \( 10! \)
Recall that \( 11! = 11 \times 10! \). We can substitute this into the equation:
\[ \frac{11}{10!} = \frac{x}{11 \times 10!} \]

### Step 5: Cancel \( 10! \) from both sides
Since \( 10! \) is present in both denominators, we can cancel it:
\[ 11 = \frac{x}{11} \]

### Step 6: Solve for \( x \)
To find \( x \), we multiply both sides by \( 11 \):
\[ x = 11 \times 11 = 121 \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{121} \]