Express each of the following in exponential notation:
` ((1)/( 24))^(13) +((1)/(24))^(16)`

Answer

To express the given expression \(\left(\frac{1}{24}\right)^{13} + \left(\frac{1}{24}\right)^{16}\) in exponential notation, we can follow these steps:

### Step 1: Rewrite the Expression
Start with the original expression:
\[ \left(\frac{1}{24}\right)^{13} + \left(\frac{1}{24}\right)^{16} \]

### Step 2: Factor Out the Common Term
Notice that \(\left(\frac{1}{24}\right)^{13}\) is a common factor. We can factor it out:
\[ \left(\frac{1}{24}\right)^{13} \left(1 + \left(\frac{1}{24}\right)^{3}\right) \]

### Step 3: Simplify the Second Term
Now, simplify \(\left(\frac{1}{24}\right)^{3}\):
\[ \left(\frac{1}{24}\right)^{3} = \frac{1}{24^3} \]
Thus, we can rewrite the expression as:
\[ \left(\frac{1}{24}\right)^{13} \left(1 + \frac{1}{24^3}\right) \]

### Step 4: Rewrite 24 in Terms of Prime Factors
Next, express \(24\) in terms of its prime factors:
\[ 24 = 2^3 \times 3^1 \]
So, we can rewrite \(24^{13}\) and \(24^{3}\):
\[ 24^{13} = (2^3 \times 3^1)^{13} = 2^{39} \times 3^{13} \]
\[ 24^{3} = (2^3 \times 3^1)^{3} = 2^{9} \times 3^{3} \]

### Step 5: Substitute Back into the Expression
Substituting these back into our expression gives:
\[ \left(\frac{1}{2^{39} \times 3^{13}}\right) \left(1 + \frac{1}{2^{9} \times 3^{3}}\right) \]

### Step 6: Combine the Terms
Now, we can combine the terms:
\[ \left(\frac{1}{2^{39} \times 3^{13}}\right) \left(\frac{2^{9} \times 3^{3} + 1}{2^{9} \times 3^{3}}\right) \]
This simplifies to:
\[ \frac{2^{9} \times 3^{3} + 1}{2^{39} \times 3^{13} \times (2^{9} \times 3^{3})} \]

### Step 7: Final Expression
Now, we can write the final expression:
\[ \frac{2^{9} \times 3^{3} + 1}{2^{48} \times 3^{16}} \]

### Final Answer
Thus, the expression \(\left(\frac{1}{24}\right)^{13} + \left(\frac{1}{24}\right)^{16}\) in exponential notation is:
\[ \frac{2^{9} \times 3^{3} + 1}{2^{48} \times 3^{16}} \]

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