If `lim_(x to a) (x^(5) + a^(5))/(x + a) = 405`, Find the value of a.

Answer

To solve the limit problem, we start with the given expression:

\[ \lim_{x \to a} \frac{x^5 + a^5}{x + a} = 405 \]

### Step 1: Substitute \(x = a\)

When we substitute \(x = a\) into the expression, we get:

\[ \frac{a^5 + a^5}{a + a} = \frac{2a^5}{2a} \]

### Step 2: Simplify the expression

The expression simplifies as follows:

\[ \frac{2a^5}{2a} = \frac{a^5}{a} = a^4 \]

### Step 3: Set the limit equal to 405

Now, we set the simplified expression equal to 405:

\[ a^4 = 405 \]

### Step 4: Solve for \(a\)

To find \(a\), we take the fourth root of both sides:

\[ a = \pm 405^{1/4} \]

### Step 5: Simplifying \(405\)

Next, we need to simplify \(405\). We can factor \(405\) as follows:

\[ 405 = 3^4 \times 5 \]

### Step 6: Calculate \(405^{1/4}\)

Now we can find \(405^{1/4}\):

\[ 405^{1/4} = (3^4 \times 5)^{1/4} = 3 \times 5^{1/4} \]

### Step 7: Write the final answer

Thus, the values of \(a\) are:

\[ a = \pm 3 \times 5^{1/4} \]