Express each of the following in expontial notation :
` (729)/( 64)`

Answer

To express \( \frac{729}{64} \) in exponential notation, we can follow these steps:

### Step 1: Factor 729
First, we need to factor 729 into its prime factors. We can do this by dividing by 3 repeatedly:
- \( 729 \div 3 = 243 \)
- \( 243 \div 3 = 81 \)
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

So, \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \).

### Step 2: Factor 64
Next, we factor 64 into its prime factors. We can do this by dividing by 2 repeatedly:
- \( 64 \div 2 = 32 \)
- \( 32 \div 2 = 16 \)
- \( 16 \div 2 = 8 \)
- \( 8 \div 2 = 4 \)
- \( 4 \div 2 = 2 \)
- \( 2 \div 2 = 1 \)

So, \( 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 \).

### Step 3: Write the fraction in exponential form
Now we can rewrite the fraction \( \frac{729}{64} \) using the exponential forms we found:
\[ \frac{729}{64} = \frac{3^6}{2^6} \]

### Step 4: Simplify using properties of exponents
Using the property of exponents that states \( \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \), we can simplify:
\[ \frac{3^6}{2^6} = \left(\frac{3}{2}\right)^6 \]

### Final Answer
Thus, the expression \( \frac{729}{64} \) in exponential notation is:
\[ \left(\frac{3}{2}\right)^6 \]

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