Express each of the following in exponential notation:
` ((-16)/( 35))^(16) + ( (-16)/( 35))^(3)`

Answer

To express the given expression \(\left(-\frac{16}{35}\right)^{16} + \left(-\frac{16}{35}\right)^{3}\) in exponential notation, we can follow these steps:

### Step 1: Rewrite the Expression
Start with the given expression:
\[ \left(-\frac{16}{35}\right)^{16} + \left(-\frac{16}{35}\right)^{3} \]

### Step 2: Apply the Law of Exponents
According to the laws of exponents, we can separate the numerator and denominator:
\[ \left(-\frac{16}{35}\right)^{16} = \frac{(-16)^{16}}{35^{16}} \quad \text{and} \quad \left(-\frac{16}{35}\right)^{3} = \frac{(-16)^{3}}{35^{3}} \]
Thus, we can rewrite the expression as:
\[ \frac{(-16)^{16}}{35^{16}} + \frac{(-16)^{3}}{35^{3}} \]

### Step 3: Find a Common Denominator
To add the two fractions, we need a common denominator. The common denominator here will be \(35^{16}\). We can convert the second term:
\[ \frac{(-16)^{3}}{35^{3}} \cdot \frac{35^{13}}{35^{13}} = \frac{(-16)^{3} \cdot 35^{13}}{35^{16}} \]

### Step 4: Combine the Fractions
Now that both fractions have the same denominator, we can combine them:
\[ \frac{(-16)^{16}}{35^{16}} + \frac{(-16)^{3} \cdot 35^{13}}{35^{16}} = \frac{(-16)^{16} + (-16)^{3} \cdot 35^{13}}{35^{16}} \]

### Step 5: Factor Out Common Terms
Next, we can factor out \((-16)^{3}\) from the numerator:
\[ = \frac{(-16)^{3} \left((-16)^{13} + 35^{13}\right)}{35^{16}} \]

### Final Expression
Thus, the expression in exponential notation is:
\[ = \frac{(-16)^{3} \left((-16)^{13} + 35^{13}\right)}{35^{16}} \]