Simplify and express each of the following as a rational number :
` ((5)/(3))^(12) + ( ( 5)/(3))^(15)`

Answer

To simplify the expression \(\left(\frac{5}{3}\right)^{12} + \left(\frac{5}{3}\right)^{15}\), we can follow these steps:

### Step 1: Factor out the common term
The first step is to factor out the common term \(\left(\frac{5}{3}\right)^{12}\) from both terms in the expression.

\[ \left(\frac{5}{3}\right)^{12} + \left(\frac{5}{3}\right)^{15} = \left(\frac{5}{3}\right)^{12} \left(1 + \left(\frac{5}{3}\right)^{3}\right) \]

### Step 2: Simplify the expression inside the parentheses
Next, we simplify \(\left(\frac{5}{3}\right)^{3}\):

\[ \left(\frac{5}{3}\right)^{3} = \frac{5^{3}}{3^{3}} = \frac{125}{27} \]

Now substitute this back into the expression:

\[ \left(\frac{5}{3}\right)^{12} \left(1 + \frac{125}{27}\right) \]

### Step 3: Combine the terms in the parentheses
To combine \(1 + \frac{125}{27}\), we need a common denominator:

\[ 1 = \frac{27}{27} \]

So,

\[ 1 + \frac{125}{27} = \frac{27 + 125}{27} = \frac{152}{27} \]

### Step 4: Substitute back into the expression
Now, we substitute this back into our expression:

\[ \left(\frac{5}{3}\right)^{12} \cdot \frac{152}{27} \]

### Step 5: Rewrite \(\left(\frac{5}{3}\right)^{12}\)
We can express \(\left(\frac{5}{3}\right)^{12}\) as:

\[ \left(\frac{5^{12}}{3^{12}}\right) \]

Thus, we have:

\[ \frac{5^{12}}{3^{12}} \cdot \frac{152}{27} \]

### Step 6: Combine the fractions
Now, we can combine the fractions:

\[ \frac{5^{12} \cdot 152}{3^{12} \cdot 27} \]

### Step 7: Calculate \(3^{12}\) and \(27\)
Since \(27 = 3^{3}\), we can rewrite the denominator:

\[ 3^{12} \cdot 27 = 3^{12} \cdot 3^{3} = 3^{15} \]

### Step 8: Final expression
Now, we can write the final expression as:

\[ \frac{5^{12} \cdot 152}{3^{15}} \]

### Step 9: Calculate \(5^{12}\)
Calculating \(5^{12}\):

\[ 5^{12} = 244140625 \]

### Step 10: Final result
Thus, the final simplified expression is:

\[ \frac{244140625 \cdot 152}{531441} \]