Find the range of x for which the binomial expansions of the following are valid .
`(x + 5)^(3//2)`

Answer

To find the range of \( x \) for which the binomial expansion of \( (x + 5)^{\frac{3}{2}} \) is valid, we can follow these steps:

### Step 1: Identify the form of the binomial expansion
The binomial expansion is valid for expressions of the form \( (a + b)^n \) where \( n \) is not a natural number. In this case, we have \( (x + 5)^{\frac{3}{2}} \).

### Step 2: Rewrite the expression
We can rewrite \( (x + 5)^{\frac{3}{2}} \) in a form that allows us to apply the binomial theorem:
\[ (x + 5)^{\frac{3}{2}} = 5^{\frac{3}{2}} \left(1 + \frac{x}{5}\right)^{\frac{3}{2}} \]

### Step 3: Apply the condition for the binomial expansion
For the binomial expansion \( (1 + u)^n \) to be valid when \( n \) is not a natural number, the condition is:
\[ |u| < 1 \]
In our case, \( u = \frac{x}{5} \). Therefore, we have:
\[ \left|\frac{x}{5}\right| < 1 \]

### Step 4: Solve the inequality
This inequality can be rewritten as:
\[ -1 < \frac{x}{5} < 1 \]
Multiplying all parts of the inequality by 5 gives:
\[ -5 < x < 5 \]

### Step 5: State the range of \( x \)
Thus, the range of \( x \) for which the binomial expansion of \( (x + 5)^{\frac{3}{2}} \) is valid is:
\[ x \in (-5, 5) \]

### Summary
The range of \( x \) for which the binomial expansion of \( (x + 5)^{\frac{3}{2}} \) is valid is \( (-5, 5) \).

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