Find the range of x for which the binomial expansions of the following are valid .
`(7 - 4x)^(-5)`

Answer

To find the range of \( x \) for which the binomial expansion of \( (7 - 4x)^{-5} \) is valid, we follow these steps:

### Step 1: Identify the form of the binomial expansion
The binomial expansion is valid for expressions of the form \( (1 + x)^n \) when \( n \) is not a natural number, provided that \( |x| < 1 \).

### Step 2: Rewrite the expression
We can rewrite the expression \( (7 - 4x)^{-5} \) in a suitable form:
\[ (7 - 4x)^{-5} = \frac{1}{(7 - 4x)^{5}} = \frac{1}{7^5} \cdot \left(1 - \frac{4x}{7}\right)^{-5} \]

### Step 3: Identify \( x \) in the binomial form
From the rewritten expression, we can see that:
\[ 1 - \frac{4x}{7} \]
is in the form \( 1 + x \) where \( x = -\frac{4x}{7} \).

### Step 4: Apply the condition for the binomial expansion
For the expansion to be valid, we need:
\[ \left| -\frac{4x}{7} \right| < 1 \]
This simplifies to:
\[ \frac{4|x|}{7} < 1 \]

### Step 5: Solve the inequality
To solve the inequality, we multiply both sides by \( 7 \):
\[ 4|x| < 7 \]
Now, divide both sides by \( 4 \):
\[ |x| < \frac{7}{4} \]

### Step 6: Write the range of \( x \)
The inequality \( |x| < \frac{7}{4} \) implies:
\[ -\frac{7}{4} < x < \frac{7}{4} \]
Thus, the range of \( x \) for which the binomial expansion is valid is:
\[ x \in \left(-\frac{7}{4}, \frac{7}{4}\right) \]

### Final Answer
The range of \( x \) for which the binomial expansion of \( (7 - 4x)^{-5} \) is valid is:
\[ x \in \left(-\frac{7}{4}, \frac{7}{4}\right) \]