Express each of the following in exponential notation:
` ((-7)/( 15))^(12) + ( ( - 7)/( 15))^(15)`

Answer

To express the given expression \(\left(-\frac{7}{15}\right)^{12} + \left(-\frac{7}{15}\right)^{15}\) in exponential notation, we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \left(-\frac{7}{15}\right)^{12} + \left(-\frac{7}{15}\right)^{15} \]

### Step 2: Apply the Exponent Rule
Using the rule \(\left(\frac{A}{B}\right)^M = \frac{A^M}{B^M}\), we can rewrite each term:
\[ \frac{(-7)^{12}}{15^{12}} + \frac{(-7)^{15}}{15^{15}} \]

### Step 3: Find a Common Denominator
The denominators are \(15^{12}\) and \(15^{15}\). To combine these fractions, we need a common denominator, which is \(15^{15}\). We can rewrite the first term:
\[ \frac{(-7)^{12}}{15^{12}} = \frac{(-7)^{12} \cdot 15^{3}}{15^{15}} \]
Now, our expression looks like this:
\[ \frac{(-7)^{12} \cdot 15^{3}}{15^{15}} + \frac{(-7)^{15}}{15^{15}} \]

### Step 4: Combine the Fractions
Now that we have a common denominator, we can combine the numerators:
\[ \frac{(-7)^{12} \cdot 15^{3} + (-7)^{15}}{15^{15}} \]

### Step 5: Factor Out Common Terms
Notice that \((-7)^{12}\) is common in both terms of the numerator:
\[ \frac{(-7)^{12} \left(15^{3} + (-7)^{3}\right)}{15^{15}} \]

### Final Expression
Thus, the expression in exponential notation is:
\[ \frac{(-7)^{12} \left(15^{3} - 7^{3}\right)}{15^{15}} \]